Development and Applications of the
Quantile Regression Estimation (QRE)
Procedure
Dale P. Lifson
lnst. of
iv
TABLE OF CONTENTS
_.e
1.
U"TRODUCTION . . . . . . . . . . . . . .
1
2.
LITERATURE REVIEW--HISTORY OF ESTIMATION IN THE 3-PARAMETER
LOGNORMAL DISTRIBUTION . . . . . . . . . . . . . . . . . .
3
3.
DEVELOPMENT OF THE QRE PROCEDURE AND DERIVATION OF ESTIMATING
EQUATIONS FOR SEVERAL APPLICATIONS . . . .
3.1
3.2
3.3
3.4
3.5
4.
APPLICATIONS OF THE QRE PROCEDURE
4.1
4.2
4.3
4.4
e
4.5
5.
The 3-Parameter Lognormal Distribution. .
The 3-Parameter Log Equicorrelated-Normal Distribution .
The SB Distribution . . . . . . . . . . . . . . . . •
The Income Distribution of Singh and Maddala . . . . .
The Conditional QRE Procedure Applied to the 3-Parameter
Lognormal Distribution . .
Monte Carlo Study I--Four 3-Parameter Lognormal
Distributions of Cohen and Whitten . . . . . .
Hill's (1963) Epidemic Data (3-parameter Lognormal
Distribution) . . . . . . . . . . . . . . . . .
Monte Carlo Study II--Six SB Distributions from Johnson
and Kotz (1970) . . . . . . . . . . . . . . . . . • .
1960-1972 U.S. Family Income Data--Income Distribution
Proposed by Singh and Maddala (1976) . . . . . . . .
Monte Carlo Study III--Application of Conditional QRE
and Ordinary QRE Procedures to Four 3-Parameter
Lognormal Distributions of Cohen and Whitten • .
SUMMARY AND CONCLUSIONS
15
16
26
33
36
41
45
46
70
76
84
92
96
LIST OF REFERENCES
100
APPENDIXES . . • .
103
APPENDIX A:
MAXIMUM LIKELIHOOD ESTIMATION IN THE 3-PARAMETER
LOGNORMAL AND SB DISTRIBUTIONS .
. . . . .
104
APPENDIX B:
SELECTED ESTIMATION ALGORITHMS .
110
APPENDIX C:
SOME INITIAL WORK ON DETERMINING THE· OPTIMAL
SPACING OF 3-PARAMETER LOGNORMAL QUANTlLES .
121
A MONTE CARLO STUDY OF THE SMALL SAMPLE DISTRIBUTION OF EXP(aZ ) . . . . . . . . . . . . . . .
126
APPENDIX D:
P
v
LIST OF TABLES
Table
Quantile regression equations for selected probability
distributions . . . . . . . .
. . . . . . . .
39
Estimated mean-squared errors (EMSE) of 14 estimation
procedures compared in Cohen and Whitten (1980)
53
Means of 3PLN parameter estimates from LMLE and QRE
procedures. .
. . . . .
59
Standard deviations of 3PLN parameter estimates from LMLE
and QRE procedures . . . . . . . . .
. . . .
60
Tests for bias of LMLE and QRE procedures in estimating the
3PLN location parameter y . . . . .
. . . .
62
Empirical relative precision CR.P.) of quantile regressionestimates to local maximum likelihood estimates of
3PLN parameters . . . . . . . .
. . . .
66
Estimated indexes of efficiency (EIE) of QRE, ME, and 12
modified estimation methods of Cohen and Whitten
68
Hill's (1963) data on incubation period of inoculated
smallpox
. . . . . . . . . ..
. . . . . .
71
4.8
Additional groupings of Hill's data into quantiles
72
4.9
Summary statistics for various 3PLN estimations of Hill
data
. . . . . . . . . .
. . . . . . . . . ..
74
Means and standard deviations of SB parameter estimates
from LMLE and QRE procedures
. . . . . . . . . ..
79
Tests for bias in SB parameter estimates from LMLE and
QRE procedures . . . . . . . .
. . . .
81
Estimated relative precision of QRE's to LMLE's for six
SB distributions
. • . • . . . . . . . .
..
82
4.13
1960-1972 U.S. family income data .
86
4.14
Parameter estimates of income distribution of Singh and
Maddala obtained from Singh and Maddala's estimation
procedure . . . . . . . . . . . . . . . . . .
87
3.1
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.10
4.11
4.12
4.15
Parameter estimates of income distribution of Singh
and Maddala obtained from weighted and ordinary QRE
procedures . . . . . . . . . . . . . . . . . . . .
~
vi
. LIST OF TABLES (Continued)
e
Table
4.16
Means and standard deviations of 3PLN parameter estimates
using conditional QRE procedure . . . . . . . . . . . . .
93
Means and standard deviations of 3PLN parameter estimates
using ordinary QRE procedure . . . . . . . . . . . . . .
95
Means and standard deviations of 3PLN parameter estimates
for various definitions of the first sample quantile (PI)
124
Values of a, p., and n used in the Monte Carlo study of
exp(aZ .) . . l.. . .
..
129
D.2
Means of exp(aZ .)
131
D.3
Variances of n ~exp (aZ .)
D.4
Skewness of exp(aZ .)
133
D.S
Kurtosis of exp(aZ .)
134
4.17
C.1
D.l
pl.
....
pl.
pJ.
pl.
pl.
•..•
132
vii
LIST OF
FIGURE~S
Figure
4.1
Probability density functions of Cohen and Whitten's
four 3-parameter lognormal distributions .
47
Variance of estimates of y vs. sample size for QRE and
LMLE procedures • . . .
64
4.3
Probability density functions of six SB distributions
77
D.l
Percentage differences between empirical and limiting
values of E[exp(aZ .)] for varyin.g values of n, p.,
1
and a . . . . . . p1. . . . . . . . . . .
. . . . . .
135
Percentage diff~rences between empirical and limiting
values of Var[n~exp(aZ .)] for varying values of n, p.,
and a . . . . . . . . P: . . . . . • . • . . . . . . : .
136
4.2
D.2
CHAPTER 1
_4t
INTRODUCTION
The
primary
regression
intent
estimation
of
(QRE)
this
work is
procedure as
to introduce the quantile
a means of estimating the
parameters of the 3-parameter lognormal distribution.
Since 1963, when
Hill uncovered' difficulties with maximum likelihood estimation of this
distribution,
gested.
several alternative estimation procedures have been sug-
These include local maximum likelihood estimation [Harter and
Moore (1966), Wingo (1975), and Cohen and Whitten (1980)], estimation by
a
regression
1970)],
on
expectations of order statistics
discrete maximum likelihood' estimation
thorne
(1976)],
[Munro and Wixley
[Giesbrecht and Kemp-
and modified maximum likelihood and modified moments
estimation [Harter and Moore (1966), and Cohen and Whitten (1980)].
The
procedure proposed here is for application to selected sample quantiles,
whereas many of the above procedures apply best to complete samples.
The QRE procedure is an extension of an idea presented in Cohen and
Whitten.
It is
so named because estimates of the parameters of an
assumed distributional
quantiles
on
their
form are obtained from a regression of sample
asymptotic
expected
values.
The
regression
is
typically nonlinear and involves iteratively reweighted least squares
estimation techniques.
While the procedure was developed in connection
with the 3-parameter lognormal distribution, it has similar applications
to
related distributions
normal
and SB
applications
their
such as the 3-parameter log equicorrelated-
(4-parameter lognormal)
distributions.
Each of these
involves approximating the expected quantile values with
asymptotic
values.
The
expected
covariance
matrix
of
the
regression residuals is also computed using the asymptotic joint distri-
2
bution of
the quantiles.
Finally,
the procedure may be used in a
somewhat more straightforward fashion to estimate. the parameters of many
continuous
probability
distributions
~1hose
inverse
~
distribution
functions are known in closed form.
The paper is organized into five ch13Lpters, including this introduction.
Chapter 2 reviews the literature on estimation of the 3-para-
meter lognormal distribution so that the proposed quantile regression
estimation procedure can be placed in peI:spective.
In Chapter 3, the
QRE procedure is developed as it applies to the 3-parameter lognormal
distribution and the necessary equations for applying the QRE procedure
to this and several other distributions a.re derived ~
In addition, a·
conditional version of the QRE procedure is developed as an alternative
means of estimating the 3-parameter lognormal distribution.
The per-
formance of the QRE procedure is empirically investigated in Chapter 4,
"
where estimation is performed on real and generated data from many of
the distributions discussed in Chapter 3.
The conditional version of
the QRE procedure is also examined empirically in Chapter 4.
and conclusions are presented in Chapter 5.
A summary
~
3
CHAPTER 2
.e
LITERATURE REVIEW--HISTORY OF ESTIMATION IN THE
3-PARAMETER LOGNORMAL DISTRIBUTION
In this chapter, the important contributions to estimation in the
3-parameter lognormal distribution are reviewed in chronological order.
The purpose of the review is to enable the reader to evaluate the
quantile regression estimation procedure proposed here in relation to
previously established estimation methods.
In the following review there are several discussions that refer to
either the density function or the likelihood function associated with
the
3-parameter
follows
meter y,
the
lognormal
distribution.
3-parameter lognormal
scale parameter IJ,
If
the
random
distribution with
and shape parameter cr,
variable X
location para-
then the density
function of X is given by
f (Xjy,lJ,cr)
X
= (2ncr2)-~(x_y)-1
for a
= 0
2
> 0,
exp{-[ln(x-y) - 1J]2 f2cr2}
(2.1)
Y < x < co
otherwise.
The logarithm of the likelihood function of a sample xl' ... , x
n
from
the 3-parameter lognormal distribution is given by
(2.2)
n
I
i=l
[In(x.-y)-IJ]
1
2
n
-
I
i=l
In(x.-y) .
1
In the review that follows, the differing notations employed by the
various authors are changed to match that of Eq. (2.1) and (2.2) above.
4
Yuan (1933).
of
the
Yuan was the first to provide a thorough accounting
properties
of
the
3-parameter
lognormal
distribution.
In
addition to deriving expre,ssions for a variety of measures of central
tendency, dispersion,
ske~"D.ess
and kurtosis, he was the first to derive
the method of moments estimators of the three parameters (y,
~
and 0).
The moment estimators are obtained by solving the following system of
equations for y,
~
and 0:
1
X
s
a
where w
2
3
= Y + w~exp(~)
= w(w-1)exp(2~)
= (W+2)(w-l)~
= exp(02) ,
and x, s 2 , and
(2.3)
8
3
are the sample mean, variance and
third standard moment, respectively.
The solution is found by first solving for w in the cubic equation
3
2
w + 3w -
(a; + 4) = O.
(2.4)
By Descartes I s "rule of signs", only one of the roots of this equation
is real, and this root will be taken as the estimate
estimators are easily defined in terms of
~
* = lnw*,
~* = In{s[W*(W*-l)]-\},
Y* = x- s(W*-l)-\ .
of w.
The other
according to
2
0
~
(2.5)
and
Wilson and Worcester (1945) and Cohen (1951).
These authors inde-
pendently developed what they thought were maximum likelihood estimators
of y,
~
and o.
However, Hill (1963) showed that the solution to the
likelihood equations corresponded to the local rather than global max-
5
imUJD.
Therefore, the estimators developed by Wilson and Worcester and
also by Cohen were actually local maximwD likelihood estimators.
logarithm
of
the
likelihood equation is
given in Eq. (2.2).
The
These
authors derived the (local) maximum likeli.hood estimators [(L)MLE's] in
the usual way,
each
of
the
setting the partial derivBltives of lnL with respect to
three
parameters
equal
to
zero
and
resulting system of equations for the par.ameters.
then
solving
the
Each easily derived
the relationships
PC)')
= (l/n)
fi(y) = (l/n)
n
I In(x.-)') and
1
i=l
(2.6)
n
[In(x.-y) - p(y)]2
I
1
i=l
so that the (L)MLE f s of IJ and C1 could be ca.lculated once the (L)MLE of y
was obtained.
Substitution of Eqs. (2.6) i.nto the equation alnLjay
= 0,
gives the expression
n
(x.-y)
I
1
i=l
-1
n
{n
n
2
In(x.-y) - n I ln (x.-y) +
I
1
1
i=l
i=l
- n
In each paper,
it was
2
n
[I In(x.-y)]2}
i=l
I [In(x.-y)j(x.-y)] =
1
recommended that
1
y
be
1
o.
(2.7)
found by employing an
iterative searching algorithm to solve the above equation for y.
Each also computed what they thought were the asymptotic variances
and
covariances
matrix.
of the
(L)MLE f s
by inve:cting the Fisher information
The validity of this procedure in the present context is in
doubt because the range of the random variable depends on the parameter
y.
The information matrix is given by
e
6
-E
[a21~
a8 a8
i
j
]
= 1(8) = I(y,J.I,o)
1
w2 (1+02)
J.l2 0 2
w~
-2w~
Ii02
IJO
w~
1J20 2
1J2 02
=
o
1
(2.8)
1
2w~
•
0
IJO
Moreover, the convergence in probability of 1(8) to 1(8) has never been
proven.
Using the notation devised by Cohen and Whitten,(l980); letA=
2
[w(l+i) - (1+20 )]-1
and C = exp(IJ)/w~.
If the probability limit of
1(8) were equal to 1(8), then the asymptotic variances and covariances
of the parameters, taken from the appropriate elements of [I(Y,J.I,o)]-l,
would be given by
e
V(y) = (02/ n )AC 2
(2.9)
V(P) = (02/ n ) (1+A)
V(o) = (02/ n ) (1+ 202A)
2
Cov(y,P) = -(0 /n)AC
and
Cov(y,o:) = (03/ n )AC
Cov(P,o) = _(03/ n )A.
Aitchison and Brown (1957).
Aitchison and Brown authored a book
devoted to the lognormal distribution which consisted of a review of
virtually
everything
including
methods
of
that
was
then
estimation.
known
about
Perhaps
contribution to parameter estimation was
their
their
method of quantiles (MQ) estimation procedure.
the
most
distribution,
significant
introduction of the
They reconunended that
7
the three quantiles X , X 5' and Xl
a.
-a be equated to their population
values and the resulting system of equations be solved to obtain the
parameter estimates.
The choice of a symmetric pattern of quantiles
permitted a
direct,
rather
equations.
Letting
c
I-a
=
than
<I>
-1
iterative solution to the system of
(1-a)
where!
<I>
is
the
inverse
of the
standard normal distribution function, the method of quantiles estimates
corresponding to X ' X. and X - are given by
a
l a
5
-1
a = cl_a[ln(Xl_a-X.S)
p = In(X.S-Xa )
- In(X . s-Xa )],
- In[l-exp(-crc
(2.10)
)] and
I-a. '
A
Y = X. S - exp(p).
Aitchison and Brown recommended that a be! set equal to . OS and noted
that the method fails if cr < O.
Hill (1963).
Hill discovered that the solution of the likelihood
equations leads to local, rather than glclbal maximum likelihood estimates.
In his
proof,
reproduced in Appemdix A, he shows that there
exists a path in the parameter space along which the likelihood of any
ordered sample xl' ... , x
approaches +co.
n
from the 3-parameter lognormal distribution
By defining estimates of 1.1 and a in terms of Y [see
Eq. (2.6)] the likelihood function is seen to approach +co as [y,P(Y;x ,
l
... , xn),a(y;x , •.. , x )]
n
1
converges to
[:x:1,-oo,+<»].
Hill states on p.
2
[x ,P(x ),cr (x )] = (xl,-OO~-tQ:l) is the
l
l
l
2
maximum likelihood estimate of (y,l.I,a ) although in fact the likelihood
75
"that in a meaningful sense
at that point is zero."
To sidestep the issue of local vs. global MLE's, Hill used Bayesian
techniques to obtain 3PLN parameter estimates.
He was able to construct
e
8
a posterior distribution for y,
·e
~,
and cr which had its greatest prob-
ability density corresponding to a y value near the local maximum of the
likelihood function rather than at Xl.
Hill's results are compared with
those from other estimation procedures in Section 4.2 of this thesis.
Lambert (1964).
Lambert
computed
LMLE' s
for 23 generated 3PLN
samples of size 32 and compared them with method of quantiles (MQ)
estimates.
His contribution was to apply the method of scoring to the
computation of LMLE' s.
partial derivatives
of
The
scoring method uses the fact that the
the likelihood function with respect to the
parameters equal zero at the point where the likelihood attains a local
maximum.
Thus,
given a· set of initial parameter' estimates,· it is
possible to iterate to the LMLE solution by successively calculating
corrected parameter estimates which satisfy second order Taylor's series
approximations
methods
to
the
of obtaining
likelihood
equations.
initial values,
identical final estimates.
Lambert proposed three
each of which lead to nearly
Strangely, he did not publish the results of
the estimation on the 23 generated samples nor did he comment on the
relative performances of the MQ and LMLE procedures.
Harter and Moore (1966).
Harter
and
Moore
investigated
local
maximum likelihood estimation in censored and uncensored samples from
the 3PLN distribution.
last
n-m
maximum
The censoring was considered for the first rand
order statistics.
likelihood
estimation
They
introduced a
procedure
for
modification 'to the
use when the
procedure led to the global rather than local maximum.
standard
They simply
censored the smallest uncensored sample value, x + ' and modified the
r 1
likelihood equations accordingly.
By using Xl as an upper bound on the
possible values of y, censoring Xl removed the problem of the likelihood
9
function becoming unbounded.
They noted that for their distribution
("FlO, J.l=4, 0=2), the modified procedure
the
They
doubly-censored samples when the
W~IS
only necessary for some of
salD]ple
size was small (n=100).
suggested that the modification might also be required for un-
censored samples of even smaller sample sizes.
In their Monte Carlo
study the modified estimates seemed to hav,e properties similar to those
of
the
conventional
LMLE's,
but
their
study
was
not
designed
to
efficiently compare the two estimation proc,edures.
Munro and Wixley (1970).
Munro and Wi:x:ley were concerned with 3PLN
parameter estimation in very small
sampl,es
(n=10,
15 and 20).
They
developed an iteratively reweighted least squares regression procedure
in which the sample order statistics were regressed on their approximate
expected values.
Eq.
(2.1)
Using a different parameterization than that given in
they were
able
to
express
statistic as a linear function of the
thE~
expectation of each order
expe~ctation
of its corresponding
standard lognormal order statistics, which depended only on the shape
parameter cr.
If the value of cr were known, the remaining two parameters could be
readily estimated as the intercept and slop,e estimates from the weighted
least-squares regression.
types of
In the more
com~on
case with cr unknown, two
approximations were required before the estimation could be
performed.
approximate
A first order Taylor's series expansion about cr was used to
the
expectations
procedure devised by Blom
of the
(1958)
was
sample
use~d
order statistics,
and a
to approximate the means,
variances, and covariances of the corresponding standard lognormal order
statistics.
e
10
Their weighted least squares procedure is similar to the quantile
regression estimation (QRE) procedure to be proposed in this thesis.
The two procedures differ in that the QRE procedure is a weighted least
squares regression on sample quantiles, rather than on the full set of
order statistics.
The use
of quantiles
permits
the
derivation of
asymptotic expectations of the first and second moments of the quantiles, thereby eliminating the need for the moment approximation techniques required in Munro and Wixley's method.
O'Neill and Wells (1972).
method
[see
Lambert
(1964)]
O'Neill
to
and
compute
Wells
(local)
used
the
maximum
scoring
likelihood
estimates of 2PLN. and 3PLN parameters from grouped samples of insurance
claim payment data.
They concluded that the scoring method worked very
well.
They then computed the efficiencies of the estimates based on two
different grouping strategies relative to those obtainable from ungrouped data.
The relative efficiency of a particular grouping was
defined as the ratio of the large sample variance of the estimate using
that grouping to the large sample variance of the estimate obtained
using ungrouped data.
These variances were computed from the inverses
of the appropriate Fisher information matrices.
The validity of their
calculations depends upon the convergence in probability of the matrix
ICe) to its true value ICe).
While this result has never been proven,
it is supported by the empirical results of Lambert (l964) and others.
For their claim payment data,
parameters
were
clearly
more
estimates of both 2PI.N and 3PLN
efficient
if
the
data
were
grouped
according to logarithmic increments rather than equal increments.
The
advantage of logarithmic grouping appeared greater when fewer groups
were formed.
11
Calitz (1973).
In this empirical study, Calitz compared the effi-
ciency of various estimation procedures i.n estimating the shape parameter a of the 3PLN distribution.
of five distributions in which
Using 50 generated samples from each
varied from 0.3 to 1.1, he computed
<1
estimates using the methods of maximum
order 5% and
95%,
and
optimal
like~lihood,
moments, quantiles of
Based on
quantiles.
the
empirical
variance of the estimates of a, Calitz concluded that the local maximum
likelihood estimator was far superior to t.he other three, and that when
a was large the method of moments was much more precise than either of
the quantile methods.
Wingo (1975).
The objective of Wingo's work was to find an algor-
ithm that permitted local maximum likelihClod parameter estimates to be
found from any 3PLN or SB 1 sample.
That is, he sought an algorithm that
would guarantee the attainment of the loc:al rather than the anomalous
global solution of the likelihood equations.
The
method
Wingo
chose
was
to
us I!
interior penalty function
techniques which transformed the constrained optimization problem (solve
for
y
subject to
y<
xl) into an unconstrained problem.
For the 3PLN
estimation, the unconstrained (transformed) objective was to maximize
the penalty function
P(Y,r) = lnL(y) - r[(y+c)
-1
+ (x1-Y)
-1
]
(2.11)
where L(y) was the likelihood function in terms of p(y) and &(y)2, c was
IThe SB distribution is also known as the 4-parameter lognormal distribution.
The density function is given in Section 3.3 of this thesis.
e
12
any large positive number (say, 10
~ 1).
25
) and r was a small number (say
The maximization of Eq. (2.11)
reductions in the value of r.
is
accompanied by successive
When r is relatively large, the second
term in Eq. (2.11) is relatively large which ensures that a solution of
y=x
1
or
~-~
will be avoided.
Once the iterative process moves in the
direction of attaining the local maximum of InL(y) the value of r is
decreased to permit the value of P(y,r) to be dominated by the value of
InL(y).
The penalty function used to estimate the two location parameters
of
the
SB
Eq. (2.11).
(4-parameter
lognormal)
distribution
was
analogous
to
Using data from Hill (1963), Tiku (1968), Lambert (1964)
and Iwai (1950), Wingo found that the penalty function approach always
converged to a reasonable solution and thus concluded that the algorithm
was robust.
Giesbrecht and Kempthorne (1976).
Giesbrecht
and
Kempthorne
pointed out that previously suggested methods of 3PLN parameter estimation have ignored the possible effects on the parameter estimates of
grouping error in the data.
within
class
intervals
of
They felt that when data are recorded
width 6 the
likelihood
function
should
properly be defined as
L(y,~,a)
2
P(y)
a(y)
=K IT p~i
(2.12)
1
=!
=!
1n(x.-y)/n,
1
[In(x.-y) - p(y)]2/n , and
1
InL(y) = -n[p(y) + lna(y)] + a constant.
13
where K is a constant, f.]. is the frequen4:y of observations in the ith
interval and p. is defined by
].
p.].
for
b
f
authors
(2.2)].
meter
(2.13)
a
= (i+.5)6.
and b
computed maximum likelihood estimates
data and for Hill's
estimates
2 -~
-1
2
2
(2no) (x-y) exp{-[ln(x-y)-~] /20 }dx
= max[(i-.5)6,y]
a
The
=
for simulated
(1963) data using Eq. (2.12) and compared them to
obtained by maximizing
the
usual
likelihood
equation
[Eq.
They found that in the case of 'the simulated data, the para-.
estimates
obtained
parameter values
using
Eq. (2.12)
were
closer
to
than were those obtained using Eq. (2.2).
the
true
Also,
it
will be seen in Chapter 4 of this thesis: that the estimates they obtained for the Hill data provided a
bett~er
fit to the data than did
those that were obtained by solVing the conventional likelihood equations.
However, estimates from the quantile regression estimation (QRE)
procedure will be seen to prOVide an even better fit to this data.
QRE
e
procedure
implicitly accounts
for
"J~rouping
error"
The
by' assigning
asymptotic expected values to the quantiles defined by the grouping.
Cohen and Whitten (1980).
Cohen and Whitten introduced two modi-
fications to the conventional maximum liklelihood and moment eS,timation
procedures.
The two modifications involvE!d replacing one of the like-
lihood equations,
or one of the moment ,equations with either of two
equations involving a small sample order statistic.
Details of their'
methodology are given in Section 4.1 of 1:his thesis.
It is concluded
there that none of their modifications
yie~ld
estimators superior to the
conventional local maximum likelihood estilliators.
Still, the ideas used
14
in the development of these modified procedures helped motivate the
development of the QRE procedure proposed in this thesis.
15
CHAPTER 3
DEVELOPMENT OF THE QRE PROCEDURE AND DERIVATION
OF ESTIMATING EQUATIONS FOR
Sl~VERAL
APPLICATIONS
In this chapter, the quantile regression estimation (QRE) procedure
is
derived as a means of estimating the parameters of the 3-parameter
lognormal
distribution,
some related distributions,
and finally,
many
continuous distributions whose inverse di.stribution functions are known
in a simple closed form.
It is hoped t.hat the relative ease of its
application will make it attractive,
especially when it is felt that
condensing the data would be prudent.
The
chapter
procedure
is
is
organized
developed
dis tribution.
Following
as
as
it
this
follo"7s.
applies
to
development,
In Section 3 . 1 ,
the
3-parameter
related
the
QRE
lognormal
applications
are
derived in Sections 3.2 and 3.3 for the 3-parameter log equicorrelatednormal
and
SB
(4-parameter
lognormal)
distributions,
respectivelY.~
These three uses of the QRE procedure rely on a technique that uses the
moment generating function of the normal random variable to derive the
asymptotic expectations of the sample quuntiles.
Then, in Section 3.4
the QRE procedure is developed in the m4)re straightforward case where
the asymptotic expectations of the quant.iles can be obtained from the
inverse of the distribution function.
Regression equations are derived
for the income distribution proposed by Singh and Maddala (1976) as an
example.
equations
Finally,
At the end of Section 3.4 a table is provided listing the QRE
that
in
pertain
Section 3.5
to
a
several
other
probability
distributions.,
conditional quantile regression estimation
procedure is derived as it applies to the 3-parameter lognormal distri- ~
16
bution.
In this procedure, the sample quantiles are regressed on their
asymptotic conditional expectations, where the conditioning variable is
the value of the preceding quantile.
3.1
The 3-Parameter Lognormal Distribution
The
distribution of the random variable X is a member of the
3-parameter lognormal family of distributions if In(X - y)
~ N(~,
2
cr ).
In this case, the density function of X is
fX(x;y,~,cr)
= (2ncr2)-~
(x - y)-l exp{-[ln(x _ y) _ ~]2/2cr2}
for cr
= 0,
2
> 0,
Y< x <
(3.1)
~,
otherwise.
The ideas leading to the development of the QRE procedure came out
of some
modified maximum likelihood
and modified moment estimation
procedures proposed by Cohen and Whitten (1980).
Their modifications
derived from the fact that the lognormal variate X is related to the
standard normal variate Z by the transformation
Z
Thus, if X
r
= [In(X
- y) - ~]/cr.
(3.2)
is the rth order statistic from a random sample from the
3-parameter lognormal (3PLN) distribution, then Z
r
= [In(Xr
- y) -
~]/cr
is the rth order statistic of a corresponding sample from the standard
normal distribution.
were
to
replace
one
The modifications proposed by Cohen and Whitten
of the
three
usual
equations
of the maximum
likelihood or moment estimation procedures with the equation
In(X r -
y) = ~ + crE(Z )
r
(3.3)
17
where X is the rth sample order statistic.'
r
tions,
alnL/ay = 0 was
replaced,
and
in the moment estimation,
equation for the third moment was replaced.
(3.3) with r
= 1,
2, and 3.
In the likelihood equathe
e
Cohen and Whitten tried Eq.
Details of their investigation are given in
Section 1 of Chapter 4.
Whereas
Cohen and Whitten' s
procedurl~s
utilized just one of the
order statistics (at a time), one might cOlJ.sider using all of the order
statistics to estimate the 3PLN parameters.
and Wixley (1970) for small samples.
This was attempted by Munro
Under a different 3PLN parameter-
ization they expressed the expectations of the 3PLN order statistics as
linear
(t
in
the
transformed
= y + I-' and u
= 0lJ),
location
and
(t)
scale
(u)
parameters
but nonlinear in the shape parameter a.
(t,u,a)
expectations involved the first two moments of standard [i.e.,
= (O,l,a)] lognormal variables and depended on the value of a.
The
However,
in addition to the problem of a being unknown, no expressions existed
for the moments of the standardized variables and therefore approximations had to be used.
Munro and Wixley used iteratively reweighted
least
to
squares
techniques
resul ts seemed to
obtain
display biased
the
parameter estimates.
estimatE~s
of
e
Their
a for sample sizes of
10, 15, and 20.
Another idea that uses the order stati;stics might be to exploit the
monotonic transformation
x
r
= y +
+ aZ ).
exp(~
r
(3.4)
If the expectation of the right-hand side of Eq. (3.4) could be computed
as
a
function of
weighted sum of [X
y,
r
~
and
a, then minimization of an appropriately
- E(X )]2 using weighted least squares techniques
r
e
18
might
-e
be a
estimates
good way
to
obtain estimates'· of the parameters.
could be obtained by
finding
The
the weighted least squares
solution to the regression equation
xr
= E(X r ) + wr
r
= 1,
(3.5)
2, ... , n
where the ware correlated residuals.
r
tional problems arise with this approach.
However, formidable computaFirst consider the expec-
tation on the right hand side of Eq. (3.5):
E(X r )
=E[y + exp(~
= y +
~exp(~).
where
E[exp(aZ r )].
+ aZ r )]
~
E[exp(aZ )],
r
A
closed-form
(3.6)
'
expression
does
not
exist
....
for
Thus to perform the regression of Eq. (3.5), either costly
and time-consuming Monte Carlo studies or other numerical integration
procedures would have to be performed to compute the empirical distributions of exp(aZ ) for numerous values of a and r, or else E(X ) and
r
r
Cov(X , X ) would have to be approximated for r,s = 1, ... ,n.
r
s
The Monte
Carlo results would be specific to the sample size, n, so that the Monte
Carlo experiments would have to be redone each time n changed.
Due to these complications the suggestion is put forth to consider
the
regression equation (3.5) with sample quantiles in place of the
sample order statistics.
When the sample size n is large the joint
distributions of the sample quantiles and the corresponding standard
normal quantiles are approximately multivariate normal.
The expecta-
tions, variances and covariances of the sample quantiles can be computed
in closed form using these limiting distributions, making the quantile
regression estimation procedure practical.
19
Since the data required for implementaltion of the QRE procedure are
sample
quantiles,
complete
set
of
these
data.
quantiles
The
must
following
sample quantiles, given in Sarhan and
first
be
dE~finitions
GrE~enberg
selected
from the
of population and
(1962), will be used.
Let g(x) be any absolutely continuous probability density function.
Definition 3.1
The pth population quantile of g(x) i.s the value X = c
J
c
p
p
such that
g(t)dt = p
-CD
Definition 3.2
The pth sample quantile of a sample of size n from density g(x) is
defined as
X = X
p
np
if np is an integer
= X([np] + 1)
if np is not an integer,
where [np] denotes the largest integer ~ np.
In order to compute the expectations required for performing quantile regressions, the limiting distributiclnal properties of the sample
quantiles must be derived.
The following theorems, taken from Sarhan
and Greenberg, specify these asymptotic distributions.
Theorem 3.1
If g(x)
is differentiable in the neighborhood of X = c
g(c ) 1 0, then the distribution of the variate
p
[n/p(1 -
p)]~
g(c ) (X
p
p
tends to N(O, 1) as n tends to infinity.
- c )
p
p
and if
•
e
20
Theorem 3.2
For k given real numbers for which
let the p.-quantile of the population be c., that is
1
1
C.
1
J
get) dt = p.;
-to
1
i
= 1,2,
... , k.
Assume that the frequency function g(x) of the population is differen~
tiable in the neighborhoods of X = c. , i = 1, ... , k, and gt
1
g(c )
i
~
O,fori=I, ... ,k.
1
Then the joint distribution of {n~(X . - c.)}, where n.=[np.]+I,
n1
i=l,
... ,
1
1
1
k tends to a k-dimensional normal distribution with zero means
and with covariance matrix
as n tends to infinity.
If Z
is the pth quantile from a random sample of n standard
p;n
normal variables then
21
lim E(Z ·n ) = c = <II -1 (p)
P'
P
n-+ClD
(3.7)
.here <II is the standard normal distribution function, and
lim n Var(Z
~here
~
p;n
)
=
p(l-p)
[ep(c ) ]2
p
(3.8)
is the standard normal probability density function.
Finally,
p(l-r)
lim n Cov(Z ,Z )
.p
~ov
r
= ep(c
p
)ep(c )'
for
p <: r .
(3.9)
r
the regression of the 3-parameter lognormal quantiles on their
asymptotic expected values may be derived, the result being closed form
expressions for the regression equation and covariance matrix.
Let X p1 ' Xp2 ' ... , Xpk be k sample qwmtiles from a random sample
of size n from the 3-parameter lognormal dis:tribution, with 0 < PI < ..•
< Pk < 1.
Let Zpl' Zp2' ... , Zpk be defined by
Z .
p1.
=
log(X .-Y)-IJ
p1.
(3.10)
a
Replacing the order statistics in Eq. (3.6) with the sample quantiles gives the regression equation
xp1.. = Y + P E[exp(aZp1..)]
Nov,
frOID Eq.
+ ~' 1.. .
(3.11)
(3.7) and (3.8) and noting that the expectation in Eq.
(3.11) is the moment generating function 4)f the asymptotically normal
variate Z ., it follows that
p1.
E[exp(OZp1..)] = exp (ac.1. + a 2 k 1.1.
.. /2) ,
(3.12)
e
22
where
p. (I-p.)
1
c. = ep -1 (p .) and k .. =
1
1
11
1
n[cjl(c.)]
2
1
Substituting Eq. (3.12) into Eq. (3.11) gives the nonlinear quantile regression equation
xp1. = Y +
~
2
exp(crc. + cr k.. /2) + w.
1
11
1
(3.13)
The variances and covariances of the errors, w., may be computed as
1
well,
using similar techniques.
The variances are computed from the
standard computation formula
Var(w.) = Var(X .)
1
p1
= E(Xp1.)2 _ [E(Xp1.)] 2 .
(3.14)
An asymptotic expression for E(X .) is given by subtracting w. from the
p1
right-hand side of Eq. (3.13).
1
E(X .)2 is computed as
p1
= E[y + ~exp(crZ .)]2
p1
= E[y2 + 2y~exp(az .) + ~2exp(2az .)]
p1
p1
= y2 + 2y~E[exp(aZ .)] + ~2E[exp(2aZ .)].
p1
p1
(3.15 )
Asymptotic expressions for the two expectations in the last line of Eq.
(3.15) may be obtained by again recognizing them as moment generating
functions of asymptotically nODDal random variables, Z ..
p1
•
2
E[exp(aZ .)] = exp(crc. + cr k .. /2),
p1
1
11
Thus,
(3.16)
and
2
E[exp(2aZ .)] = exp(2ac. + 20 k .. )
p1
1
11
(3.17)
23
Substituting Eq. (3.16) and (3.17) into Eq.
(3~15)
gives
Subtracting the square of E(X .) in Eq. (3.13) from Eq. (3.18) gives the
p1
desired expression for the asymptotic variance of the sample quantiles:
Var(X . ) ; ~2exp(2ac.)exp(a2k.. )[exp«r2k .. ) - 1].
p1
1
11
11
(3.19)
The covariance between any two quantiles X . and X . is computed
p1
PJ
similarly, as follows.
Cov (X ., X .) = E(X . X .) - E (X r)E (X .).
p1
PJ
p1 PJ
P1"
PJ
(3.20)
The first expectation in Eq. (3.20) is computed as
(3.21)
E (X . X .) = E{[ Y + ~exp (aZ .)][ y + ~exp (aZ .)]}
p1 PJ
p1
PJ
2
y + YPE[exp(aZ .) + exp(aZ .)] + ~2E{exp[a(Z . + Z .)]}.
p1
PJ
p1
PJ
=
In the last term, Z . + Z . is a linear combination of asymptotically
p1
PJ
normal, correlated random variables.
e
Thus,
Z . + Z . .:. N( c. + c., k.. + k.. + 2k. .:1 ,
p1
PJ
1
J
11
JJ
1J
(3.22)
where
k ..
1J
Again,
= [p.(l-p.)]/[n~(c.)~(c.)]
1
J
1
J
making use
for p. < p ..
1
J
of the normal moment :generating function and Eq.
(3.22) gives
2
E{exp[a(Z . + Z .)]}; exp[a(c. + c.) + a (k .. + k .. + 2k .. )/2].
p1
PJ
1
J
11
JJ
1J
Substituting Eq. (3.16) and Eq. (3.23) into Eq. (3.21) gives
(3.23) .
24
.·.e
E(X .X .) ; ~2 + ~p{exp[cr(c. + c.)]
p1
PJ
(3.24)
J
1
2
2
+ exp [ (cr k.. + cr k .. ) 12]}
JJ
11
2
2
P exp{cr(c.1 + c.)
+ cr [(k .. +
J
11
+
k .. + 2k . . )/2]).
JJ
1J
Expanding the product of the expectations in Eq. (3.20) gives
(3.25)
•
2
2
E(X .)E(X .) = [y + pexp(crc. + cr k.. /2)][y + pexp (crc. + cr k .. 12) ]
p1
PJ
1
=~
2
J
11
JJ
+ yp{exp[cr(c. + c.)]
J
1
2
2
+ exp [ (cr k.. + cr k .. ) 12]}
JJ
11
+
2
2
P exp[cr(c.1 + c.)
+ cr (k .. +
J
11
k .. )/2].
JJ
Finally, subtracting Eq. (3.25) from Eq. (3.24) gives
Cov(X ., X .)
p1
PJ
=• p2 {exp[cr(c.1
X
2
+ c.)]exp[o (k .. + k . . )/2]
J
11
JJ
(3.26)
[exp(cr2k .. ) - I]).
1J
Defining, in matrix form for a given parameter vector
e=(~, ~,
0),
(3.27)
w(6) = x - u(6), and
V(6) = E {[wee)] [w(6)]'},
the solution vector 6 =
(y, p,
&)
is that vector .e for which the
corresponding weighted sum of squares of the errors
[WSSE(e)] is a
minimum, where
WSSE(6) = [x - u(6)]' [Vee) ]-1 [x - u(e) 1
.
(3.28)
25
Since
depends
the asymptotic expression for
on the parameters
the covariance matrix V(8)
being esti.mated,
(8)
(3.28) must be found via an iterative prc)cess.
the solution to Eq.
The estimation algo-
rithm, described in detail in Section 1 of Appendix B, initially sets
V(8)
equal to the identity matrix and uses the one-step Gauss-Newton
procedure to compute an approximate ordinary least squares estimate of
e.
This estimate of 8 is used to compute a new estimate of V(8) which
is then used in the one-step Gauss-Newton procedure to re-estimate 8
using weighted least squares.
vee)
e
The process 4)f recomputing an estimate of
and obtaining the corresponding weighted least squares estimate of
continues
until
convergence
in
the
parameters
is
attained.
No
guarantees that the algorithm will converge to the solution vector 8
have been discovered, although the
algorit~D
performed very for well for
the data herein.
The QRE procedure described above depelClds upon the first and second
order moments
of the distributions
of the sample quantiles being in
close agreement with the corresponding momelClts of the asymptotic distributions of the quantiles.
The rate of convlergence of the sample. moments
to their asymptotic expectations was examilCled in a Monte Carlo experiment reported in Appendix
that of exp (aZ .).
p1
D.
The relevant distribution to examine was
This was done for various values of a and p., and
for various sample sizes n.
1
The results of the experiment show close
agreement between the sample moments and thleir limiting expectations for
most values of a and Pi when n is as small
,ilS
59.
e
26
·e
3.2
The 3-Parameter Log Equicorrelated-Normal Distribution
... ,
Z
be n+l independent standard normal random
n
variables and consider the transformation
Z~ = p~Z 0 + (l-p)~Z.; i=l, 2, ... , n
~
(3.29a)
~
for some p in the interval (0,1) .. Also, let Y , ... , Y be independent
1
n
standard normal random variables and let Yo be distributed N(O, 1) but
1
1
with E(YoY.) = -(-p)~/(l-p)~, and consider the transformation
~
(3.29b)
for some p in the interval [-(n-l)-I,O).
Each of these transformations induces the formation of n equicorrelated
N(O, 1)
random
variables.
pertains to the case where p > 0.
The
discussion
The random variables
that
Z~
1.
follows
may be shown
to be equicorrelated and N(O,I) distributed as follows:
E(Z~)
= E[p~Z + (I-P)~Z.]
1.
0
1.
=
p~(Z o )
+
(I-P)~(Z.)
=
1.
(3.30a)
°,
E(Z~)2 = E[p~Z + (I-P)~Z.]2
1.
0
(3.30b)
1.
= E[ pZ2 + 2p~(I-p)~Z Z. + (l-p)Z~]
001.
+
= p • 1
1.
+ (l-p) • 1 = 1 ,
0
and
E(Z~Z~) = E{[p~Z + (I-P)~Z.] [p~Z + (l-p)~Z.l}
1.J
0
1.
0
J
(3.30c)
27
= E[ pZ 2 + p~(l-p)~Z (Z. + Z.) + (l-p)Z.Z.]
o
01,
J
1.J
2
= pE(ZO)
+ p~(l-p)~[E(2; Z.) + E(Z Z.)] + (l-p)E(Z.Z.)
=p
+
1
1
o
• 1
1.
0
J
o
1.
J
+ 0
=p
Now consider a sample X., i=l, ... , n where the X. are defined by
1.
1.
the transformation
log(X. -y) - IJ
J.
=
a
(3.31)
Z~
1.
The random variables X. will be said to be distributed as 3-parameter
1.
log equicorrelated-normal (3PLEN) random "ariables.
The QRE procedure
is applicable in this equicorrelated situ.ation, as will now be shown.
Just as in the uncorrelated case, in the equicorrelated case, Eq.
(3.20)
represents a monotonic transformat.ion between the X. and the
1.
original, uncorrelated Z..
Indeed for any two values x. and x.
1.
x. <
1.
X.
J
-+
1.
log(x.-y) < log(x.-y)
1.
log(x·-Y)-IJ
1.
a
z! <
1.
1
-+
-+
P ~z
o
z. <
1.
(3.32)
J
-+ --,..;;;,..--
-+
J
log(x.-Y)-IJ
<
J
(since a > 0)
z~
J
+ (l-p) \z. < P~2,
1.
0
+ (l-p) ~z.
J
1
Z.
J
[since (1 - p)~ > 0] .
Therefore the rth order statistic X can be related to the rth standard
r
normal order statistic Z by
r
e
28
log(X r -)')-1-'
(3.33)
cr
or equivalently
1
1
X
r
=)'
+ exp[1-' + crp~z
0
+ cr(l-p) ~Z
r
l.
(3.34)
It follows then, that the same relationship exists between the two sets
of quantiles:
X . = y + exp[1-' + crp ~ Z+ cr(I-p) ~ Z
o
p~
.l.
(3.35)
pl.
Thus, the quantile regression equation
Xp~.
= E(Xpl..)
(3.36)
+ w.
~
can be estimated provided that closed-form expressions for E(X .) and
p~
E(w w') can be computed, where w is the vector (wI' ... , Wk)'.
By making use of Theorems 3.1 and 3.2 and again using the expectation given by the normal moment generating function, these expectations may be computed with respect to the asymptotic distributions of
the quantiles, giving
(3.37)
and
E(w w')
=
Var(X )
p1
Cov(X ,X )
p1 p2
Cov(XpI,Xpk )
Cov(X ,X )
p1 p2
Var(X )
p2
Cov(X ,X )
p2 pk
------- ------Cov(Xp1,Xpk )
Cov(X ,X )
p2 pk
------Var(X )
pk
29
where
(3.38)
tit
and
(3.39)
·2
2p)exp[a(l-p) ~ (c.+c.)]exp[a2 (l-p)(k .. +k .. )/2]
exp(a
Cov(X . ,X .)=~
p~
PJ
~
J'
~~
JJ
2
2
x {exp(a p)exp[a (1-p)k .. ] - I}.
~J
The
asymptotic variances and
covariances
given above are derived as
,.;
follows
Var(X .) = E(X .)2 _ [E(X .)]2
p~
p~
(3.40)
p~
= E[y+~exp(aZ' .)]2
p~
_ {E[y+~exp(aZ' .)]}2
p1
= E[y2 + 2y~exp(aZ'.) + ~2exp(2aZ' .)]
4
p:L
p~
_ {y + ~E[exp(aZ' .)]}2
p~
= y2
+
2y~E[exp(aZ'p~.)] + ~2E[exp(2(J'Z'p~.)]
_ {y2 + 2y~E[exp(aZ' .)] + ~~2[exp(az'.)]},
p~
.
p~
which simplifies to
Var(X .)
p~
2 ·
= ~2
{E [exp(2aZ' .)] - E [exp(aZ' .)]},
p~
After using Eq.
p~
(3.41)
(3.29) to substitute for :Z'., the expectations in Eq.
pI.
(3.41) may be derived as
E[exp(2aZ' .)]
pI.
= E{exp[2ap~Z 0 + 2a(l·.p)~ZpI..]}
= E{exp[2ap~Z o ]exp[2a(l-p)~ZpI..]}
(3.42)
= E{exp[2ap\Zo]}E{exp[2a(l-P)~Zpi]}'
the last step following due to the independence of Z and each Z .,
o
pI.
In
e
30
the last line of Eq.
(3.42) the first expectation is the moment gen1
erating function of the normal random variable 2ap~Z.
o
Thus,
(3.43a)
The second expectation in the last line of Eq. (3.42) may be approximated by treating it as the moment generating function of the asymptotically normal random variable 2a(1-p)~Z ., giving
p~
= exp[2a(1-p)~c.]exp[2a2 (1-p)k .. ].
~.
E{exp[2a(1-p) Z .]}
P~
1
~
(3.43b)
~~
Substituting Eqs. (3.43) into Eq. (3.42) gives
(3.44)
E[exp(2aZ' .)]
P~
=• exp(2a2 p)exp[2a(1-p) ~c.]exp[2a2 (l-p)k .. ].
~
~~
Taking the second expectation on the right-hand side of Eq. (3.41) gives
1
2
E2
[exp(aZ'.)]
E {exp[ap~Z
P~
=
2
0
+ a(l-p) ~ Z .]}
(3.45)
P~
~
1
= E {exp[ap~Z ]exp[a(1-p) Z .]}
o
P~
•
2
1
2
= exp(a p/2)exp[a(1-p)~c.]exp[a (l-p)k .. /2].
~
Finally,
substituting Eqs.
~~
(3.44) and (3.45) into Eq.
(3.41) gives
P2 exp(a2 p)exp[2a(1-p) \ c.]exp[a2 (l-p)k .. ]
Var(X .)
p~
~
X
2
~~
(3.46)
2
{exp(a p)exp[a (l-p)k .. ] - I}.
~~
Similar methodology is used to derive the covariances.
Cov(X ., X .) = E(X .X .) - E(X .)E(X .)
P~
PJ
P~ PJ
p~
PJ
E{[y + pexp(aZ' .)][y + pexp(aZ' .)]}
p~
PJ
- E[y + pexp(aZ' .)]E[y + pexp(aZ' .)]
=
P~
PJ
(3.47)'
31
= y 2 + ypE[exp(aZ'p1.)]
+ E[e~)(aZ' .)]
.
PJ
+ p~[exp(aZ' .)exp(aZ' .)]
p1
PJ
- y2 _ ypfE [exp(aZ' .) + E [l~xp(aZ' .)]}
p1
PJ
- P~ [exp (aZ'p1.)] E [ exp (aZ'PJ". ') ]
= p2 {E[exp(OZ'.
p1
Substituting Eq.
Cov(X . ,X .) =
p1 PJ
(3.29) into this last explC'ession and simplifying gives
p2{E[exp(2ap~Z )]E[exp(a(l.-p/~(Z
0
1
- E[exp(ap~Z
The
+ aZ'.)] - E[exp(aZ' .)]E[exp(aZ'.)].
PJ
p1
PJ
1
(3.48)
~
1
+ a(l-p)~Z .)]E[exp(ap~Z + a(l-p) Z .)]}.
o
p1
0
PJ
expectations in Eq.
four
. + Z .»]
p1
PJ
(3.48)
necessary, the expectations are
are
computed as
approximatE~d
follows.
When
as moment generating func-
tions of normal random variables.
E[exp(2ap~Zo )]
= exp(2a2 p).
(3.49a)
E{exp[a(l-p)~(Z .+ Z .)]}
(3.49b)
p1
PJ
2
; exp[a(l-P);(C. + c.) + a (:L - p)( k.. + k.. + 2k .. ) /2] .
1
J
11
JJ
1J
1
E{exp[ap~Z
1
o
+ a(l-p)~Z .]}
(3.49c)
p1
2
•
~?
= exp(2a p)exp[a(l-p) c. + a"·(l-p)k .. /2].
1
E{exp[ap;Z
o
11
+ a(l-p)~Z .]}
(3.49d)
PJ
=. exp(2a2p)exp [a(l-p) ~?
c. + a"· (l-p)k .. /2].
J
JJ
Finally, substituting Eqs. (3.49) into Eq. (3.48) and
Cov(X .,X .)
p1 PJ
·22
=
P exp(a p)exp[a(l-p) ~ (c.
1
+ c.)]
J
simplifyi~g
gives
(3.50)
2
2
x exp[a2 (1-P)(k .. + k .. )/2]{exp(a p)exp[a (1-P)k .. ] -I}.
11
JJ
1J
.
In theory,
fashion
section;
as
that
the
3PLEN parameters
described
for
could be estimated in the same
the 3PLN parameters
in the previous
i.e., by employing an iteratively reweighted generalized non-
e
32
linear least
squares
algorithm.
One necessary modification of the
procedure would be to restrict the estimate of p to lie between zero and
one.
This procedure was tried for 3PLEN samples' generated from distri-
butions with p=O.5 and (Y,IJ,o) corresponding to each of the four 3PLN
distributions of Cohen and Whitten (see previous section).
In these
analyses, parameter estimates could not be obtained because the matrix
of derivatives of the regression equation with respect to the parameters
was nearly singular.
Thus, the Gauss-Newton algorithm broke down when
an attempt was made to invert this nearly singular matrix.
However, the
estimation worked well when p was fixed at its true value.
promising alternative
estimation procedure would be
further iteration on fixed values of p.
Perhaps a
to introduce a
33
3.3
The SB Distribution
The QRE procedure may also be readily applied to the SB distri-
bution
(sometimes
referred to
as the 4-parameter lognormal distribu-
tion).
The SB distribution was first described by Johnson (1940) as one
of a trio of distributional families that together spanned all possible
combinations
parameter
of
skewness
and
kurtosis.
lognormal distribution,
As
in
the
case of the 3-
an SB distributed random variable X
can be transformed easily to the standard :normal random variable Z.
In
this case,. the transformation has the form
=
Z
~((X-a)/(~-X)~
for
a
a > 0, a < X <
~.
(3.51 )
The possible values of an SB random variable X are bounded below by a
and above by~.
The parameter ~ determines the skewness.
the density function
the
sign of
the
i~
sYmmetric, otherwise the sign of
skewness.
Finally,
kurtosis, which decreases as
the parameter
a increases.
If ~ is zero,
~
determines
a determines the
The density
functi~n
of X is
given by
2 -~
= (2na)
(~- a)((x 2
x exp-(1/2a ){In((x for
a)(~
- x)] -1
a)/(~
- x)]
a > 0, a < x <
- ~} 2 ,
(3.52)
~,
= 0, otherwise.
(Further information about the SB family is given in Johnson (1949), and
Johnson and Kotz (1970).]
34
The QRE procedure cannot be applied to the SB distribution unless
the SB quantiles X . are related to the standard normal quantiles Z . by
p1
.
p1
a monotonic transformation.
Defining Y
Y
Since Eq.
monotonic.
This may be proven as follows.
= In[(X-a)/(~-X)],
= IJ
Eq. (3.51) can be rewritten as
(3.53)
+ aZ.
is linear in Z, the transformation from Y to Z is
(3.53)
Thus if Y can be shown to be a monotonic transformation of
of X, it will follow that Z is also a monotonic transformation of X.
The transformation from X to Y is monotonic since, for a < x. < x. <
1
J
x. < x. '+ x.
a < x.
a
1
1
J
J
-
-
x.-a
1
x.-a
< -l~-x.
1
'+--
~-x.
1
x. -a )
~~xi
'+
In (
'+
y. < y.
1
~,
x.-a
< --l~-x.
J
( x. -a )
~~Xj
< In
J
Since X is a monotonic transformation of Z, the quantiles Y . and
p1
Zp1. are related by
Y .
p1
= IJ + aZ p1..
(3.54)
The corresponding quantile regression equation is
Y .
p1
= E (IJ
+aZ .) + w. •
p1
This formulation does not allow the range parameters, a and
est~ated
the
directly.
est~ation
If a and
(3.55)
1
~
are known there is no problem.
~
to be
If not,
may proceed iteratively, via a grid-search in the (a,
~)
35
plane
for
weighted
values.
that
[a, P, pea, P), or(a, P)]
quartet
sum of squared deviations
of the X.
p1
that
from
minimizes
the
their predicted
An algorithm to perform this type e)f estimation is described in
Section 3 of Appendix B.
When ex and
P
are held constant, the following expressions define
the asymptotic expectations necessary to p,erform the regression of Eq.
(3.55) :
(3.56a)
E (fJ + crZ .) = fJ + crE(Z .)
p1
p1
= fJ + a4>
-1
(p.) ,
1
2
Var(w. ) = cr k .. , and.
1
11
2
Cov(w. ,w.) = cr k .. ,
]. J
1J
.(3.56b)
.
where $
-1
(3.56c)
is the standard normal distribution function, and k .. and k ..
11
1J
denote the asymptotic variances and covar:Lances,
respectively, of the
standard normal quantiles.
A small Monte Carlo study is given in section 4.3 in which the
sampling behavior of quantile regression estimates is compared to that
of local maximum likelihood estimates for
in Johnson and Kotz.
~le
six SB distributions given
A proof that the sollltion to the likelihood equa-
tions results in local maximum likelihood l:!stimates, rather than global
maximum likelihood estimates, is presented in Appendix A.
36
3.4
The Income Distribution of Singh and Maddala
In the three previous sections of this chapter, the QRE procedure
has been used in connection with density functions whose integrals do
not exist in closed form.
The QRE procedure is much easier to apply
when the distribution function is explicitly known and its inverse is
expressible
in
a
concise
form.
The purpose of this
section is
to
present the methodology for applying the QRE procedure in such cases.
Then,
as an illustration, the QRE equations pertaining to the income
distribution proposed
Estimation
of
this
by
Singh and Maddala
distribution
using
U.S.
(1976)
family
will be derived.
income
data
is
presented in Section 4.4 of Chapter 4.
Let xl' .•. , x
n
be a random sample of size n from a continuous dis-
tribution with density fCx) and distribution function F(x).
Also, let
< ... < x pk be k sample quantiles and c < ••• < c be k corresk
1
-1
ponding population quantiles, where 0 < PI < ••• < Pk < 1. If F
is a
X
p1
concise mathematical expression, and if the first two sample moments of
the
quantiles
of
X
converge
to
the
corresponding moments' of their
limiting multivariate normal distribution, then the QRE procedure may be
In such cases, E(X .)
applied straightforwardly.
p1
=• F -1 (p.)
= c.1
1
giving
the quantile regression equation
Xp1.
=E(Xp1.)
+ w.
1
= c.1
+ w.
1
(3.57a)
where
ECw.) = 0,
(3.57b)
1
.
2
Var(w. ) = [p.(l-p.)]/[n f (c.)],
1
111
and Cov(w. ,w.) = [p.(l-p.)]/[n f(c.)f(c.)].
1
J
1
J
1
J
(3.57c)
(3.57d)
37
Application to Income Distribution of Singh. and Maddala
In their 1976 paper, Singh and Maddala proposed the following
e
distribution function for income (X):
=1
FX(x)
b -c
- (1 + ax)
x>
,
0, a > 0, c >
o.
(3.58)
Upon differentiating Eq. (3.58), the density of X is found to be
= [abcx (b-I) ]/[(1
f X (x)
b (c-I)
+ ax )
].
(3.59)
The inverse of the distribution function is derived by solving the
distribution function for X as follows:
~
~
~
~
(3.60)
FX(x) = p
b -c
1 - (I + ax)
= p
(l + axb)-c = 1 - P
b
1 + ax
b
ax
~
~
=
o - p)-O/c)
= [(1 - p)-(I/C) - 1]
b
x
= (I/a)[(l - p)-(I/C) - 1]
x
= {(l/a)[(1 _ p)-(l/C) _ 1]}(I/b)
= F
Defining a*
= l/a,
b*
-1
(p).
= I/b
and c*
= l/c,
gives
F -1 (p) = (a*[(1 - p) -c*
(3.61)
Thus, from Eq. (3. 57a) the quantile regresl;ion equation for the "SinghMaddala" distribution is
Xp1.
= (a*[(I-p.) -c*
1
- I]}
b*
+ w..
1
(3.62)
The variances and covariances of the errors are computed according to
Eqs. (3.57c) and (3.57d) where f(x) is given by Eq. (3.59).
tit
38
In summary, this straightforward applicaton of the QRE procedure
involves
a
regression of sample
quantiles
on their expected values,
where the functional form for the expectations is the inverse of the
distribution function.
type
of
regression is
distributions,
Thus,
the only independent variable for this
the vector of quantile points, p ..
~
the quantile regression is nonlinear.
For many
Examples of dis-
tributions for which the regression is linear are the uniform, logistic,
Gumbel and exponential.
The quantile regression equations corresponding
to these and several other well-known density functions are listed in
Table 3.1.
Since
the
covariance structure always
depends
on the estimated
parameters of the density function [see Eqs. (3. 57c) and (3 .57d)] the
estimation
should proceed recursively,
i. e.,
the parameters and the
covariance matrix should be estimated in an alternating fashion until
the parameter estimates from successive iterations differ by less than a
desired convergence criterion.
such
an
solution.
estimation
In general, there is no guarantee that
algorithm will produce the global least squares
For most distributions there will be a neighborhood around
the solution vector within which parameter start-values will lead to the
desired least squares solution.
.,
j
Table 3.1.
QuanLile Regression EquaLionH for SelecLed ProbabiliLy DisLribuLlolls
Distribution
f(x)
-1
Uniform
(b-a)
Logistic
-(x-a)/b
h[l:e-(x-a)/b]2
Pareto
(2-parameter)
I[a,b](x)
I(_~,~)(x)
a xa
0
a+1 lex ~)(x)
X
0'
=. F-1 (p)
F(x)
E(X )
[(x-a)/(b-a)]I[a,b](x)
a + (b-a)p
[1 + e-(x-a)/b]-l I
(x)
( -~,~)
a - b In[(l-p)/p]
a
[1 - (x oIx) lI( x ,CDlex)
x / (l-p) l/a
P
0
0
for a > 1
Pareto
(3-parameter)
Gumbel
ax o
(x+b)a+1
I
(xo'~)(x)
b -1 ye -y for b > 0,
(x-a)/b
where y e
I (-~,~) (x)
=
Exponential
-a
a e
.
-ax I (O,~ lex)
[1 - xo
(x+b)x
lI( , CD) (x)
o
(ax o /p) 1/ (a+l) - b
e- Y
a - b In(-ln p)
where y=e
(1 - e
(x-a)/b
-ax
I(_~,~)
( )
)I(O,~) x
( )
x
a
-1
In (l-p)
-1
w
-------------------------------------,.------------------------\0
e
e
e
e
e
e
Table 3.1.
Distribution
f(x)
Truncated
Exponential
[1-e
Weibull
abx
-aT
]
-1
ae
(Conlinued)
F(x)
-ax I
( )
(O,T) x
a[l-e
~
-aT -1
-ax
]
[a(l-e
)]
E(Xp )
=• F-1 (p)
a -1 In[l-pa -2 (l-e -aT-l
)]
x I(O,T)(x)
b-l
b
exp(-ax )I(O,~)(x)
3bce -cx (I-be -cx )-(I-b) 3
von Bertalanffy
1-(l-b) 3
x
I(O,~)(x)
for b, c >
b
[1-exp(-ax )]I(O,~)(x)
[-(I/a)ln(l-p)]I/b
(l_be- cx )3_(I_b)3
1-(I-b)3
I(O,~~x)
-c1 1n
1
b{I-[P-(P+l)(I-b)3]1/3}
o.
~
o
41
3.5
The Conditional QRE Procedure Applied to the 3-Parameter Lognormal
~
~
Distribution
Let X ' X '
p1
p2
... , X be k quantiles from a sample of size n from
pk
a 3-parameter lognormal distribution.
That is, In(X - y) ~ N(~,a2).
In
Section 3.1 it was shown that
E(X .)
pJ.
=Y+
where c. = $
J.
exp(~
-1
2
+ ac. + a k .. /2)
J.
(3.63)
J.J.
p. (l-p.)
(p.) and k .. = -J.- - -J.2 .
1
1J.
n[C!>(c.)]
J.
Expressions for the asymptotic values of Var(X .) and Cov(X ., X .) were
pJ.
also
given,
p1
PJ
thus permitting the generali.zed nonlinear least squares
estimation of the regression equation
xpJ..
= E (XpJ..)
+
W.
J.
where w. is an error term.
1
(3.64)
For some applications the conditioning of the covariance matrix of
the quantiles may be too poor to permit meaningful regression results to
be obtained.
In these instances, one may wish to obtain unweighted, or
ordinary QRE t s.
A second alternative, for which the covariance matrix
is diagonal is proposed here.
The approach will be called "conditional
quantile regression estimation" (CQRE).
The CQRE approach regresses the sample quantiles on their asymptotic expectations conditional on the values of the preceding quantiles.
The reason for adopting a conditional regression is to reduce the need
for generalized nonlinear least squares by incorporating the correlation
among the quantiles into the regression model itself.
It is hoped that
e
42
by absorbing the correlations among the quantiles into the regression
equation,
ordinary
nonlinear
least
squares
techniques will provide
estimates nearly as precise as those given by the generalized nonlinear
least squares regression of Eq. (3.64).
The conditional QRE application to the 3PLN distribution is derived
as follows.
The regression model is
xp~.
= E(X
.) + u.,
~
i
p~
= E(Xpi I X
=1
. l'Xp,~• 2""'Xp 1) + u.,
~
p,~-
(3.65)
i
= 2,
3, ... , k.
Since the quantiles of a sample form a Markov chain, the regression
Eq. (3.65) can be written in an equivalent, simpler form as
X .
p~
The u.
1.
·1
= E (Xp~
X
. 1) + u ..
p,~-
1
are uncorrelated residuals since, for any sequence of random
variables {X.},
the transformed random variables Y.=X.-E(x.lx.
1""'X )
1.
1
1
1
11
are independent.
The conditional expectation in this regression may be approximated
by treating the associated pairs of standard normal quantiles as having
<.
a bivariate normal (BVN) distribution.
Defining Z . = [In(X .-y)-~]/a,
p1
p1
Theorem 3.2 says that
(Z ., Z .) :.. BVN (c. ,. k .. , c., k .. , p .. )
p~
PJ
where p..
~J
~
~~
= k .. / (k..
~J
~~
J
JJ
(3.66)
~J
k .. ) ~
JJ
With this result, the asymptotic expression for E(X·.
p1
as follows:
I XPJ.)
is computed
43
E (X.
p1
X.)
PJ
= E [y
+ exp (lJ + aZ .)
p1
I XPJ.]
(3.67)
= Y + exp(lJ)E[exp(aZ .)
I xPJ.]
=Y +
I;~
p1
exp(lJ)E [exp(aZ .)
p1
•
.],
PJ
the last step following because Z . is a monotonic transformation of X .
PJ
PJ
and therefore knowledge of X • is equivalent to knowledge of Z .. Now,
PJ
PJ
E[exp(aZ .)
p1
2
.] ; exp[aE(Z ., Z .) + .5a Var(Z ., Z .)]
PJ
p1
PJ
p1
PJ
rZ
(3.68)
since the expectation is the moment generat.ing function of the asymptotically normal random variable Z . conditil:mal on Z ..
p1
PJ
expectations on the
right-h~nd
side of Eq.
The conditional
(3.68), for large samples,
are given by
E(Z ., Z .) ; c. + p.. (k .. /k .. )~(Z ..• c.)
p1
PJ
1
1J 11 JJ
PJ
J
(3.69)
= c. + a .. (Z . - c.) wherte a .. = p .. (k .. /k .. )~
1
1J
PJ
J
1J
1J
11
JJ
and
= k .. (1 - p .. 2) = p.. (say).
11
1J
(3.70)
1J
e
Substituting Eq. (3.69) and (3.70) into Eq. (3.68) and then substituting
Eq. (3.68) into Eq. (3.67) gives
E (X .
p1
I
(3.71)
X .) = y + exp [lJ +
PJ
ac. + aa . . (:Z . - c.) + . sa2 p. .] •
1
1J
PJ
J
1J
Eq. (3.71) still cannot be used in the est.imation of the CQRE equation
because the Z . are unknown.
Substituting In(X .-Y) - lJ for aZ . takes
PJ
PJ
PJ
care of this problem and gives, after rearr,anging,
E(X ., X .) = y + exp[(l - a .. )lJ + (c. - a . . c.)a +
P~
PJ
1J
~
1J J
a ..
x (X . _ y) 1J .
PJ
.Sp . .a 2 ]
1J
(3.72)
e
44
Eq. (3.66) may now be estimated by ordinary or· weighted least squares by
replacing E(X
.1
p1
j=i-l).
X . 1) with the right hand side of Eq. (3.72)
p1-
(with
For weighted least squares estimation, the weight matrix would
be diagonal since the u. are uncorrelated.
1
A Monte
Carlo
study of
the sampling behavior of 3PLN parameter
estimates given by the CQRE procedure using ordinary least squares is
presented in Section 4.5.
The study is formatted identically to that
used
properties
to
regression
investigate
the
estimation
of
the
3PLN
of
the
unconditional
distribution
provide an additional means of comparison,
quantile
in Section 4.1.
To
Section 4.5 also includes
estimation results of the unweighted (or ordinary) QRE procedure applied
to the same generated data.
QRE procedure
covariance
The latter procedures is identical to the
except that the
matrix
identity matrix.
Vee)
is
known asymptotic form of the residual
replaced by the
These estimation results were obtained because the
ordinary QRE procedure would be
procedure
for
appropriately dimensioned
the
logical competitor of the CQRE
investigators preferring not to use the weighted least
squares techniques.
45
CHAPTER 4
APPLICATIONS OF THE QRE: PROCEDURE
In 'this chapter , five empirical analy:Sles are performed which should
permit assessments of the performance of the QRE procedure in many of
the
applications
discussed
in Chapter 3.
In each case at least one
other procedure is applied to the data so' that the QRE results may be
evaluated on'a relative basis.
The most thorough of the following allalyses is that pertaining to
the 3-parameter lognormal (3PLN) distribut.ion.
For this distribution a
large-scale Monte Carlo study is presented in Section 4.1 in which
quantile regression estimates are compared with local maximum likelihood
est:ilIlates for four 3PLN distributions and over a wide range of sample
sizes.
Then, in section 4.2, the QRE pI:ocedure is applied to Hill's
(1963) epidemic data and the results are compared to those obtained by
Hill and others who used different estimation methods.
A smaller Monte Carlo study, presented in Section 4.3, compares
quantile
regression estimates and local maximum likelihood estimates
from each of six SB distributions depicted in Johnson and Kotz (1970).
,Then,
illustrating
a
more
straightforward
application
of
the
QRE
procedure, the income distribution proposE!d by Singh and Maddala (1976)
is
est:ilIlated
in Section 4.4 using United States family income data.
These results are compared with those of a simpler estimation procedure
used by Singh and Maddala.
Finally, the conditional QRE procedure is investigated in Section
4.5 using the same generated 3PLN data uSl!d in Section 4.1.
Unweighted
QRE results are also computed for these data and each set of results is
compared with the weighted QRE results of Section 4.1.
e
46
4.1
Monte Carlo Study I--Four 3-Parameter Lognormal Distributions of
. Cohen and Whitten (1980)
In chapter 3 the theory was developed to allow the estimation of
the parameters of the 3-parameter lognormal (3PLN) distribution via a
regression of selected sample quantiles on their asymptotic expected
values.
Now the sampling behavior of the
esti~ates
given by this quan-
tile regression estimation (QRE) procedure will be examined in a Monte
Carlo experiment.
Cohen and Whitten (1980) have already conducted a Monte Carlo study
to compare the sampling behavior of 3PLN estimates from 14 estimation
procedures.
They chose four 3PLN distributions from which to generate
random samples.
Each of the four 3PLN random variables studied had a
mean of zero and a variance of one, but the skewness, a , was varied
3
from low (<<3
= 0.301)
to high (<<3
= 2.475)~.
The probability density
functions of these four random variables are plotted in Figure 4.1.
These same four distributions will also be employed in this Monte Carlo
study to facilitate comparisons of the performance of the QRE,procedure
with those of the 14 estimation procedures studied by Cohen and Whitten.
Before discussing the details of the present Monte Carlo study it
will be useful to review the design of Cohen and Whitten's study, describe the 14 methods they investigated and assess the results they
obtained.
Once this has been done, the design of the present study will
be outlined, and its objectives stated.
~The four values
Then the results will be pre-
for skewness (<<3) corresponded to setting w
2
1.10, 1.25, and 1.50 where w exp(a ).
=
= 1.01,
G--IO.OO, H-2.296, S-.0998 I
M- I . 104,/1 5-.3087
f G--3. 16,
0.8
~
'-'
'H
0.7
O.St
,
,........,.
0.4
~
'H
0.6
0.5
0.4
0.3
0.2
O. I
-2
0
x
2
AI
0.0
6
-2
G--2.00, M-.5616, 5-.4724
-2
0
4
6
X
G--1.41, M-.1436,/1 5-.6366
0.61
-
-
~
'-'
~
"!"t
'H
'H
I
0.7
0.6
0.5
0.4
0.3
0.2
O. t
-2
Figure 4.1.
o
X
2
.4
O. 0
e
WL-.L..oLL-.l..--L--.l~:I::II__""'~-J
x
Probability density functions of Cohen and Whitten's four )-parameter lognormal distributions.
~
......
e
e·
e
48
sented .and discussed in light of these objectives.·
Finally, general
conclusions will be drawn concerning the usefulness of the QRE procedure
as a means of estimating the parameters of the 3-parameter lognormal
distribution.
A Review of Cohen and Whitten's Monte Carlo Study
Cohen and Whitten conducted their Monte Carlo study in an effort to
assess the properties of twelve different estimation methods proposed in
their paper as alternatives to the moment estimation (ME) and local
maximum
likelihood estimation
(LMLE)
procedures.
Improvements
were
sought for the former because of the well-documented evidence of the
inefficiency of its estimates [see Aitchison and Brown (1957) or Calitz
(1973)] and for the latter because of uncertainty about the behavior of
LMLE's compared to that of conventional maximum likelihood estimates
(MLE' s).
The twelve modified estimation methods of Cohen and Whitten consist
of three variations each of two basic modifications of the ME and LMLE
procedures.
Recall that the traditional ME's and MLE's are obtained via
the simultaneous solution of m equations where m is the number of unknown parameters (m=3 in the case of the 3PLN distribution).
Both of
the modifications proposed by Cohen and Whitten called for the replacement of one of the three equations with an aUXiliary equation involVing
a small order statistic. ~
The same two types of auxiliary equations
were used to modify both the ME and HIE procedures.
~In
the case of LMLE, the equation alnL/ay=O was replaced. In the case
of ME, the equation of the third sample moment to its expectation was
replaced.
49
The first modification corresponded to the auxiliary equation that
equated the expectation of the cumulative distribution function of the
rth sample order statistic to its observed value, i.e.,
(4.1)
E(F(X )) = F(X ).
r
r
Cohen and Whitten point out that F(X r )=$(Z
. r ) where Zr =[In(X r -Y)-J.l]jo,
and
<I>
is the standard normal distributicln function.
Defining k r
=
<1>-1 [r/(n+1)] and using E[F(X )] = r/(n+l), Eq. (4.1) may be reexpressed
r
as
(4.2)
In(X r -Y) = J.I + ok r .
The second modification corresponded to the auxiliary equation in
which k of Eq. (4.2) is replaced by E(Z ), the expectation of the rth
r
r
standard
normal
order
statistic.
Therefore, the auxiliary equation
corresponding to Cohen and Whitten's second modification is
(4.3)
In(X r -Y) = J.I + crE(Z r ).
Values of E(Z) were obtained in tables provided by Harter (1961).
r
Cohen and Whitten investigated the performance of these two modifications using each of the first three sample order statistics (i.e.,
r = 1, 2 and 3).
Thus, their twelve modified methods consisted of two
modifications [Eq. (4.2) and Eq. (4.3)] using three order statistics in
turn and applied to two estimation methods (ME and MLE).
As mentioned earlier,
the
twelve modifications
along with the
conventional moment and local maximum likEdihood estimation procedures
were applied to samples generated from fc:>ur different 3PLN distributions.
Each of the estimation procedures was applied to 100 indepen-
50
dently generated samples of size 100 from each of the four distributions.
In generating the samples, the sample skewness was checked and
if it fell outside the interval 0.13 to 14.0 the sample was discarded.
This was done to avoid known difficulties in applying the method of
moments procedure to samples of very high or low skewness.
Indeed, when
the sample skewness is negative, moment estimators do not even exist.~
Cohen
and
Whitten's
algorithms
for
their
modified
estimation
methods involved iterative searching routines that did not always converge.
For some samples, the search for the estimate of the location
parameter y diverged toward -ao or converged to the first sample order
statistic, Xl'
When this occurred the estimation procedure was termin-
ated and was said to have failed for that sample.
The
tables,
results of Cohen and Whitten I s study were presented in two
The first contained the means of the parameter estimates over
the successful applications of each method.
The second contained the
corresponding standard deviations of these estimates.
The information
in these
to be
according
tables
to
permits the
three
14 estimation methods
important
criteria:
accuracy,
compared
precision
and
applicability.
The accuracy of a procedure may be investigated by comparing the
means of its parameter estimates to their true values.
The precision is
estimated by the standard deviations of the parameter estimates, while
~A negative
sample skewness will yield moment estimates identical to
those given by a sample with a positive skewness of the same magnitude.
However, as Cohen and Whitten point out, such estimates are inadmissible because there are no negatively skewed 3PLN distributions,
51
the applicability of a procedure will be dl!fined here as the proportion
of the time that the estimate of y does Ilot diverge toward
-00
or Xl·
e
When all three criteria are considered jointly it will be seen that
the unmodified LMLE procedure outperformedl the ME and twelve modified
procedures.
their
This is not the conclusion dn.wn by Cohen and Whitten, but
evaluation
of the results
systematically conducted.
did not
appear to be carefully or
The simple, but informative analysis of their
results given below will illustrate the superiority of the performance
of the LMLE method.
Perhaps the most popular empirical me.:lns of comparing the relative
goodness of different estimators is to use the mean-squared error criterion.
This criterion permits the accura1cy and precision of the esti-
= t(Xl ,
... , X ) is an
n
then the mean-squared err()r (MSE) of T
is defined as
mators to be jointly evaluated.
estimator of
e,
Formally, if T
e
e
(4.4)
e
Here, the mean-squared error criterioll will be used to compare the
goodness of the fourteen procedures' estim,ates of y, since this is the
parameter for which it is most critical to obtain good estimates.
To
use this criterion empirically, the expectation on the right-hand side
of Eq. (4.4) must by approximated.
where
y.1.
is the estimate given by T
An obvilous choice is given by
y for thle ith sample.
The right hand
side of Eq. (4.5) may be reexpressed as
(l/n)
~
i
(Y i
- y)2
=
(4.6)
e
53
Table 4.1
Estimated Mean-Squared Errors (EMSE) of 14 Estimation
Procedures Compared in Cohen and Whitten (1980)
Estimation Distribution 1 Distribution 2
Procedure
a = .980
a = .301
3
3
(T )*
EMSE
Rank
EMSE
Rank
Y
ME
MME-I
MME-I
1
2
MME-I
3
MME-II
MME-II
MME-II
1
LMLE
MMLE-I
e
MMLE-I
32.92
3
5.236
9
1.1218
10
.4419
10
531.41
14
11.524
11
.4194
8
.0399
7
160.79
10
15.184
12
.2462
6
.0386
6
206.98
11
6.312
10
.3057
7
.0467
8
271.03
12
1.727
1
.1623
3
.0214
2
71.66
7
2.236
7
.1338
2
.0244
3
131.41
9
61.195
14
.1859
4
.0328
5
21.35
1
1.931
4
.1165
1
.0168
1
274.18
13
26.597
13
.7054
9
.0597
9
118.58
8
1.913
3
-t
61. 73
5
2.012
5
28.75
2
2.955
8
61. 79
6
1.884
2
61.73
4
2.012
6
2
3
1
2
MMLE-I
3
MMLE-II
MMLE-II
MMLE-II
*
1
2
3
MME:
LMLE:
MMLE:
-lor -II
t
.2384
5
.0289
method of moments
modified method of moments
local maximum likelihood estimation
modified maximum likelihood estimation
ME:
r
Distribution 3 Distribution 4
a = 1.625
. a3=2.475
3
Rank
EMSE
EMSE Rank
r
first or second modification type using the rth order
statistics where r
1, 2, or 3
=
a dash (-) signifies that the estimation algorithm failed to
converge for at least 50 percent of the samples.
4
54
the modification.
For the modified maximum likelihood estimation algor-
ithms, this was true only if the first order statistic was used.
When
the second or third was used, convergence fcliled often, the failure rate
increasing with the ske\\''Oess of the
underl~ring
distribution from around
25 percent for the third most skewed distribution to well over 90 percent for the most skewed distribution.
For the least skewed distri-
bution, the failure rate varied between 8 and 27 percent across the 12
modified estimation methods.
Cohen and Whitten recommended liME-III (with MME-I
natives) when skewness is less than one.
they recommended MMLE-II
1
(with MMLE-I
l
1
and ME as alter-
For skewness greater than one,
and LMLEas alternatives).
It
is difficult to agree with their recommenda1:ions based on the results in
Table 4.1.
Perhaps they felt that their modifications had other advan-
tages over the LMLE procedure, such as ease of implementation.
The remainder of this
section will be devoted to assessing the
performance of the quantile regression
es~timation
(QRE) procedure in
estimating the parameters of the same four 3PLN distributions used by
Cohen and Whitten.
LMLE estimates will also be computed to provide a
convenient reference for use in evaluating the goodness of the quantile
regression estimates.
Monte Carlo Study Comparing QRE and LMLE Pr()cedures
In the Monte Carlo experiment conductl~d for the present study, the
sampling behavior of estimates given by the QRE and LMLE procedures were
compared.
The format of Cohen and Whitten's Monte Carlo study was du-
plicated in that 100 samples were generated from each of the four 3PLN
distributions
shown
in
Figure
(4.1).
However
in this
experiment,
samples were not discarded if the skewness lay outside the interval 0.13
55
to 14.0.
Rather, all samples were retained regardless of skewness so
that unbiased estimates of the means and variances of the parameter
estimates
could
be
obtained.
Also,
the
scope
of
this
study was
increased beyond Cohen and Whitten's in that the properties of the
estimators were investigated over a wide range of sample sizes.
An important consideration in investigating the behavior of quan-
tile regression estimates for varying sample sizes is the coordination
of the choices of the quantile points and the sample sizes.
Since the
vector of chosen quantile points p = (PI' P2' ... , Pk) has an influence
on the properties of the QRE's, the same choice of p should be used for
all sample sizes.
This avoids confounding the effects on the behavior
of the QRE' s of sample size and quantile selections.
But once p has
been chosen, only a limited set of sample sizes will contain quantiles
that correspond to p.
Therefore, in this study the following criterion
for sample size selection was used:
Necessary Condition for Sample Size Selection n:
Given a vector of quantile points P = (PI' P2' .•. Pk)' n is an
acceptable sample size for the Monte Carlo study if for each i = 1,
2,
... ,
k there exists an integer r. such that r./(n+l) = p ..
1.
1
1
Satisfying this criterion ensures that for each quantile point Pi
the sample will contain an order statistic X . for which
r1.
r./(n+l) =p.,
1
1
and therefore the p. th quantile X . is defined by X . = X ..
1.
p1.
pl.
rl.
present study, all p. were chosen to be multiples of 0.01.
1
(4.7) "
In the
In this
"".
56
case, all sample sizes one less than multiples of 100 would satisfy the
necessary condition mentioned above.
Spe1c:ifically, the following ten
quantile points were chosen:
p = (.03, .07, .15, .25, .35, .50, .65, .80, .90, .98).
These were chosen somewhat arbitrarily although an effort was made to
space them fairly evenly and to capture information near the extremes of
the data, especially at the low end.
Further consideration of how the
quantiles might be spaced is given in Appendix C.
When comparing the performance of the QRE and LMLE procedures it
must be kept in mind that only 10 quantiles are used to generate a quantile regression estimate while the entire sample is used to generate a
local
maximum
likelihood
estimate.
Under
the
assumption that
the
properties of the LMLE's are nearly the same as those normally possessed
by MLE' s~ (consistent and asymptotically efficient), the QRE's cannot be
expected to be more accurate or to have
LMLE's.
smaller variances than the
e
Rather, the objectives of this Monte Carlo study are to assess
how quickly the bias of QRE estimates di.sappears as sample size increases,
and
to
estimate
their precisi10n relative
to
the
LMLE' s.
Additionally, any failures of the algorithms to converge will be noted.
The lognormal samples of size n were created by generating n independent standard normal values zl' z2' ... , zn and transforming them to
~This assumption about the behavior of the LMLE's is supported by the
results of Cohen and Whitten's Monte Caz:lo study, where the sampling
variances of the LMLE's were shown to compare similarly to their asymptotic expectations given by the invers«~ of the Fisher information
matrix. A very small bias was indicated in the estimates of y when the
skewness was less than one, but this could have been partially due to
the elimination of samples having skewness of less than 0.13.
e
57
lognormal
.•. x
values
n
by
the
3-parameter
lognormal
trans-
formation
x.
1
=y +
exp(~
(4.8)
+ oz.).
1
For a given sample size the same 100 samples of standard normal variates
were
used
samples.
in
the
formation of each of the four
This was
done
to
sets
of lognormal
improve the efficiency of comparisons of
parameter estimates across the four distributions.
The estimation algorithms used to obtain the QRE's and LMLE's are
documented
quantile
in
Section 1
regressions,
required.
of
Appendix B.
start values
for
To
the
nonlinear
the parameter estimates were
Two separate computer programs were written which differed
only in the method of obtaining start values.
putes
perform
pseudo-moment estimators
The first program com-
(described in Appendix B)
for use as
start values, and thUS, requires positively skewed samples.
The second
program, intended only for use in the Monte Carlo study, uses the true
parameter values as start values and thus does not require that samples
be positively skewed.
values were used as
positively skewed
Convergence was slightly faster when the true
start values,
samples
but the parameter estimates
were always
the
same
regardless
from
of which
method of start value selection was used.
In searching for
the local maximum likelihood estimates" it is
important to have knowledge of the shape of the likelihood function.
The shape of the likelihood function of a typical 3-parameter lognormal
sample has been discussed in Hill (1963) and Griffiths (1980).
Unless
skewness and sample size are both small, the likelihood function as a
function of
y
should have a local maximum followed by a minimum and then
58
y=
approach += at
local maximum has
of finding the
sol~ng
the
likelihood
(y,p,o)
solution
Much of the motivation for searching for methods
xl.
come Clut of a concern that merely
equations
= (x ,-Cl:I,oo).
1
would
yield
the
undesirable
MLE
Fortunately', using the true values of y
as start values in the searching algoritrua used here always led to the
local maximum rather than to the inadmis:sible solution.
samples
from the
least skewed 3PLN dist:ribution, both the QRE' sand
LMLE's of y occasionally diverged toward
sample size (n=99).
negative
estimates
However, for
especially for the smallest
-0),
In actual applications of these procedures, large
of y should
suggest to the investigator that the
sample. may not have come from a lognormal population.
dist~ribution,
it may be better to assume a symmetric
In these cases, .
such as the normal.
In this study, the estimation algorithms of the QRE and LMLE procedures
were terminated if the estimate of y fell below -50.
whE~n
For the least skewed distribution,
the sample size was 99, the
estimation of y was terminated for 15 of 100 samples using the LMLE
procedure and for 17 of 100 samples using the QRE procedure.
For larger
sample sizes, the estimation algorithms heLd to be terminated for fewer
than 10 percent of the samples.
Among the total of 1500 samples from
the three more skewed distributions, the QFffi estimation algorithm had to
be terminated only once, for a sample of size 99, while the LMLE algorithm converged for all samples.
The
mean
procedure
estimates,
are
parameter
shown
e.g.,
estimates,
in Table
")
s.d. (y
Y = n -1"
Iy.,
:lI:
e.g.:,
4.2.
The
~
standard deviations
~ 2,
= ( n-1) -1"
I(y.-y)
~
given by each
are
of the
given in Table 4.3.
Each table also indicates, under the coluums labeled "Runs", the number
of samples (out of 100) for which the estimation algorithm converged to
59
Table 4.2.
Means of 3PLN Parameter Estimates from
LMLE and QRE Procedures
Method = LMLE
Sample Size
n
Distribution 1
99
299
499
699
899
Distribution 2
e
99
299
499
699
899
Distribution 3
99
299
499
699
899
Distribution 4
99
299
499
699
899
y
cr
1.1
Method = QRE
Runs
y
cr
1.1
Runs
(y = -10, 1.1 = 2.2976, cr= .0998, skewness = 0.301)
-10.943
-13.273
-10.698
-10.926
-10.533
2.190
2.414
2.283
2.318
2.300
.1297
.1006
.1076
.1029
.1035
85
98
98
99
100
-11.431
-13.181
-11.092
-11.398,
-11.504
2.172
2.390
2.317
2.327
2.367
(y = -3.1623, 1.1 = 1. 1036, cr = .3087, skewness
-3.249
-3.216
-3.106
-3.144
-3.118
1.093
1.108
1.077
1.091
1.083
(y = -2, 1.1
-1. 981
-1. 998
-1. 963
-1. 982
-1. 973
= .5816,
.5568
.5759
.5557
.5671
.5615
(y = -1.4142, 1.1
-1. 393
-1.409
-1.395
-1.406
-1. 403
.3236
.3089
.3191
.3129
.3147
.1174
.1376
.1223
.1337
.1295
.4924
.4743
.4860
.4779
.4801
100
100
100
100
100
1.127
1.135
1.103
1.108
1.107
83
92
96
97
100
= 0.980)
.3283
.3077
.3147
.3110
.3092
99
100
100
100
100
cr = .4724, skewness = 1.625)
100
100
100
100
100
= . 1438,
.6616
.6383
.6513
.6407
.6426
-3.423
-3.334
-3.192
-3.209
-3.194
.1414
.1073
.1065
.1046
.0987
100
100
100
100
100
-2.079
-2.041
-2.000
-2.008
-2.004
cr
= .6368,
-1.440
-1.428
-1.412
-1.416
-1.414
.5988
.5966
.5768
.5812
.5797
.4929
.4725
.4809
.4754
.4742
100
100
100
100
100
skewness = 2.475)
.1547
.1532
.1375
.1419
.1394
.6614
.6380
.6478
.6406
.6397
100
100
100
100
100
60
Table 4.3.
Standard Deviations of 3PLN Parameter Estimates from
LMLE and QRE Procedures*
Method
Sample Size
n
Distribution 1
99
299
499
699
899
Distribution 2
99
299
499
699
899
Distribution 3
99
299
499
699
899
Distribution 4
99
299
499
699
899
y
(y
IJ
y
.6057
.5520
.3718
.3533
.3073
.0665
.0489
.0338
.0324
.0290
skewness
9.662
8.777
5.103
6.303
5.033
85
98
98
99
100
.2768
.1681
.1131
.1058
.0927
.0815
.0524
.0375
.0337
.0286
100
100
100
100
100
1.2684
.7481
.4147
.4598
.3102
= -2, IJ = .5816, = .4724,
(j
.1752
.1104
.0740
.0673
.0613
.0826
.0539
.0390
.0340
.0284
100
100
100
100
100
.4653
.2405
.1449
.1550
.1137
(j
.1354
.0846
.0567
.0485
.0451
.0842
.0541
.0388
.0334
.0260
100
100
100
100
100
(j
.1820
.1040
.0633
.0672
.0510
Runs
= 0.301)
.6828
.5961
.3968
.4174
.3601
skewness
.3361
.2236
.1367
.1418
.1082
skewness
= -1.4142, IJ = .1438, = .6368,
.1161
.0679
.0468
.0415
.0348
=QRE
IJ
= -3.1623, IJ = 1.1036, a = .3087,
.2837
.1673
.1146
.1049
.0896
(y
Runs
(J
(J
.9558
.4988
.3334
.3087
.2580
(y
Method
= -10, IJ = 2.2976, = .0998,
7.758
9.103
5.480
4.536
3.583
(y
= LMLE
.0779
.0591
.0394
.0362
.0323
83
92
96
97
100
,
= 0.980)
.0978
.0678
.0452
.0421
.0328
99
100
100
100
100
e
= 1.625)
.2356
.1459
.0893
.0915
.0739
skewness
.1792
.1139
.0699
.0699
.0591
.1049
.0698
.0472
.0435
.0335
100
100
100
100
100
= 2.475)
.1084
.0724
.0496
.0452
.0345
100
100
100
100
100
*Table entries are the standard deviations of the associated set of parameter estimates. For example, for Distribution 1, the 85 estimates of
y had a standard deviation of 7.758.
61
an estimate of y greater than -50.
inferences
from
the
additional
analyses
raw results
of
these
It is difficult to draw precise
of these
results.
tables without performing
Casual
inspection
results does reveal some broad findings, however.
of
these
First, both methods
tended to estimate the parameters fairly accurately, even for sample
sizes as small as 99.
Second, as expected, the variances of the QRE's
were nearly always greater than those of the LMLE's.
Some further manipulations of the results shown in Tables 4.2 and
4.3 reveal more detailed comparisons.
The accuracy of the procedures
over the various sample sizes and distributions is ,examined in Table
4.4, with regards to the estimates of the location parameter y.
This
table shows the percentage differences between the mean estimates and
the true values of y, the t-statistics corresponding to these differences, and the probability of a greater absolute value of t assuming the
estimators are unbiased.
An index of the relative precision of the QRE
procedure compared to the LMLE procedure in estimating the 3PLN parameters will also be discussed.
Looking
first at the tests for bias in Table 4.4, note that in
general, the percentage differences between y and the mean estimates of
y
from both procedures were very small.
Over the three most skewed
distributions, the means of the LMLE's of y were different from the true
value
of y by less than three percent.
For small sample sizes,
the
estimated bias of the QRE's tended to be greater than that of the LMLE's
over these same. three distributions,
percent.
but
it was
For the least skewed distribution,
still less than 10
the mean estimates of y
from both procedures ranged from 5 to 33 percent less than the true
value of y.
These relatively large absolute differences from the true
62
Table 4.4.
Tests for Bias of LMLE and QRE Procedures in Estimating
the 3PLN Location Pu'ameter y
Method
Sample Size
n
99
299
499
699
899
(y
(y
99
299
499
699
899
(y
99
299
499
699
899
0.95
0.09
1.83
0.92
1.35
Distribution 4
(y
99
299
499,
699
899
= (y -
IJ
1.1
= .0998,
= 1.1036 ~
.3620
.2799
.0882
.5429
.0877
= .5816,
CJ
0.67
0.10
3.20
1. 76
3.02
1.1
skewness
-14.31
-31.81
-10.92
-13.98
-15.04
.2624
.0004
.2073
.0422
.1368
-0.91
-1.08
1. 70
0.61
1. 71
= -1.4142,
1.52
0.35
1.34
0.56
0.80
where
t
= -3.1623,
CJ
or
= .3087,
.5011
.9140
.0014
.0778
.0025
= .1438,
1.85
0.73
4.05
1.92
3.23
-3.97
-2.05
-0.01
-0.41
-0.21
CJ
= .6368,
.0640
.4650
<.0001
.0542
.0012
= 0.980)
-1. 71
-1. 71
-0.01
-0.52
-0.38
the mean of the y's.
t = t-statistic corresponding to H : E(y) = y.
o
.0881
.0881
.9893
.6005
.7075
= 2.475)
-1.44
-1.34
0.42
-0.25 '
0.06
y)/y; i.e., the percentage difference between
y is
.0406
.0219
.4805
.3135
.3106
1.625)
skewness
-1.85
-0.98
0.19
-0.12
0.02
.1772
.0005
.0361
.0289
.0028
-2.05
-2.29
-0.71
-1.01
-1.01
::;
I
= 0.301)
skewnes~
skewness
Prob>1 t
-1.35
-3.48
-2.10
-2.19
-2.99
-8.25
-5.42
-0.93
-1.47
-0.99
= .4724,
= QRE
tt
6(y,y)*
prob>ltl
-1.12
-3.56
-1.26
-2.03
-1.49
= -2,
Method
IJ - 2.2976,
-2.76
-1. 70
1.80
0.59
1.39
Distribution 3
6(y,y)
= -10,
-9.43
-32.73
-6.98
-9.26
-5.33
Distribution 2
*
tt
6(y,y)*
Distribution 1
= LMLE
y and
.1495
.1806
.6714
.8004
.9521
y,
e
63
= 299
value of y were significant at the 5 percent level for n
699 among the LMLE' s and for n
~
299 among the QRE' s.
and n
=
This apparent
negative bias may be explained by the fact that the skewness of the
empirical distribution of
y from each procedure was negative.
Indeed,
while the true value of y was -10, the estimates ranged from -49.5 to
-2.9.
Such a highly skewed distribution of estimates will cause the
mean of the distribution to be smaller; hence, the negative bias.
On the whole, however, the potential for bias in either estimation
procedure for n
~
99 seemed to pose little cause for concern.
For any
given sample from a 3PLN distribution having low skewness, the error
that occurs in estimating y is likely to be more
ance of
y
dures.
y
to the high vari-
than to any bias in the estimation procedure.
deviations of
variance of
due
y
y shown
The standard
in Table 4.3 reveal that when skewness is low, the
is disappointingly high for both the LMLE and QRE proce-
For samples from more skewed 3PLN distributions, the variance of
becomes much smaller, but so does the estimated bias of
y.
While
several of the tests for bias of the LMLE's in Table 4.4 reject unbiasedness, the percentage differences between the mean of
y
and yare
small enough to be ignored for most purposes (less than 2 percent).
Since bias for sample sizes greater than 99 is not a problem in
either procedure, the precision of the two procedures becomes the important point of comparison.
Figure 4.2 plots the standard deviations of
the estimates of y from each procedure as a function of the sample size
n.
The four panels shown correspond to the four 3PLN distributions.
Between each pair of curves is a shaded region indicating the difference
in the estimated precision of the two procedures.
When the differences
are a small percentage of the standard deviations of the LMLE' s, the
64
SIGMA-.099B
SIGMA-.30B7
<>-
~
~
0
~
tJ
d
al
""'o
SO
10
~
tJ·
d
~
CIS
s..
"1"4
s..
co
c:3
>
>
to
00
99
299
499
SB
9
99
299
N
•
499
•
SB
•
!!B
N
SIGMA-.4724
SI(~I1A- •6368
LMlE0Cl)-I--....,..----.....-_ _-
99
....
899
00+--...,....
93
299
. - -...._---..
493
f!B
arB
N
Figure 4.2. Variance of estimates of y vs. sample size for QRE and'
LMLE procedures.
65
relative precision of the QRE procedure to the LMLE procedure is high.
Figure 4.2 reveals that the percentage differences in precision are not
that large, and remain fairly constant for sample sizes between 99 and
899.
In addition to graphically comparing the precision of each method,
the plots· reveal that the standard errors of
y rapidly
decrease as the
sample size decreases or as the skewness increases.
Another useful means of comparing the precisions is to form an
index of relative precision CR.P.).
of the standard deviations.
=
R.p.(a)
This was done by forming the ratio
That is, for a given parameter a,
s.d. ca)LMLE
x.
s. d. Ca)QRE
where s.d.
= standard
deviation.
Since the true variance of QRE's is
larger than that of LMLE's, the population mean of the ratio R.P. must
lie between zero and one.
The R.P. estimates, shown in Table 4.5, indicate that over a wide
range of a-values and sample sizes, the QRE procedure estimated the
location parameter y with at least 60 percent of the precision of the
LMLE procedure.
The relative precision does not seem to depend very
much on the sample size for n
~
99, but does seem to depend on the
skewness of the parent distribution.
tive
to
the
LMLE' s
average values
tended
The precision of the QRE's rela-
to decrease as skewness increased.
The
(over all sample sizes) of R.P. (y) for example, were
0.86, 0.75, 0.71, and 0.67, respectively, across the four distributions.
Also, the QRE procedure compared more favorably to the LMLE procedure in terms of the precision of the estimates of I.l and a than in
terms of the estimates of y.
Estimates of R.P. averaged 0.806 and 0.813
66
Table 4.5.
Empirical Relative Precision (l~.P.) of Quantile Regression
Estimates to Local Maximum Likelihood Estimates
of 3PLN Paramet«~rs*
Sample Size
n
Distribution 1
R.P.(y)
(y
= -10,
IJ
99
299
499
699
899
a= . 0998, skewness :: 0.301)
0.8871
0.9261
0.9370
0.8465
0.8533
0.8029
1. 0371
1.0737
0.7196
0.7120
Distribution 2
(y
= -3.1623,
99
299
499
699
899
IJ :: 1.1036,
0.7535
0.6667
0.8039
0.6714
0.8316
Distribution 3
(y
= -2,
0.6098
0.6954
0.7910
0.6764
0.7877
Distribution 4
(y
(1
= .3087,
0.8541
0.8282
0.8587
0.8962
0.8957
skewness
0.8234
0.7S18
0.8275
0.7458
0.8564
IJ :: .5816, a= .4724, skewness
99
299
499
699
899
= -1.4142, IJ = .1438,
99
299
499
699
899
*
= 2.2976,
R.P.(a)
R.P·(lJ)
0.6380
0.6530
0.7386
0.6173
0.6820
0.8331
0.7721
0.8313
0.8000
0.8733
= 1.625)
0.7437
0.7570
0.8283
0.7359
0.8296
a= .6368, skewness
0.7556
0.7430
0.8112
0.6937
0.7640
= 0.980)
0.7875
0.7725
0.8268
0.7825
0.8502
= 2.475)
0.7773
0.7464
0.7819
0.7394
0.7532
Empirical relative precision of QRE proc:edure to LMLE procedure with
respect to estimates of a parameter 6 is defined as
R.P. (6)
=
s.d. (6)LMLE
x
s.d. (6)QRE
where s.d. = standard deviation.
e
67
for JJ and a, respectively, but only 0.748 for y (estimates taken over
all sample sizes and a-values).
Clearly, there is not a great sacrifice to be made in estimating
3PLN parameters from grouped, rather than raw data.
loss in efficiency of up to 40 percent.
Still, there is a
It is hoped that the informa-
tion in Table 4.5 will be helpful to those faced with the decision of
whether or not to condense large datasets.
Thus far, the QRE results have been compared only to LMLE results.
Since the Monte Carlo study presented here was patterned after that of
Cohen and Whitten, in which fourteen estimation methods (including LMLE)
were compared, it is sensible to compare the present QRE results with
the results of their thirteen alternatives to the LMLE procedure.
The
mean-squared error criterion has already been used to make comparisons
among the methods presented in Cohen and Whitten (see Table 4.1).
These
comparisons presented strong evidence for the superiority of the LMLE
procedure to the other thirteen in estimating the location parameter y.
Now, the estimated mean-squared error (MSE) of the QRE procedure
will be computed and compared with those of Cohen and Whitten's methods.
To facilitate the interpretations of the results, the estimated MSE's of
the QRE and thirteen other procedures will be divided-into the estimated
MSE of the LMLE procedure to form estimated indices of efficiency (EIE).
Thus,
higher
estimated
indices
of
efficiency
correspond
to
more
efficient estimation procedures according to the MSE criterion.
The results of the computations of the estimated indexes of efficiency are shown in Table 4.6.
The denominator of each index was the
estimated MSE for Cohen and Whitten's LMLE results rather than for the
LMLE results from the present study.
In this way, all estimates of MSE
68
Estimated Indexes of Efficiency (EIE) of QRE, ME, and 12
Modified Estimation Methods of Cohen and Whitten.
Table 4.6.
Distribution
1
2
3
4
QRE
.22
1.15
.52
.50
ME
.65
.37
.10
.04
.04
.17
.28
.42
.13
.13
.47
.44
.10
.31
.38
.36
1
.08
1.12
.72
.79
2
.30
.86
.87
.69
.16
.03
.63
.51
.08
.07
.28
.18
1.01
.17
t
.35
.96
.74
.65
.49
.58
.35
1.02,
.35
.96
Procedure*
MME-I
MME-I
1
2
MME-I
3
MME-II
-
MME-II
MME-II
MMLE-I
MMLE-I
3
1
2
MMLE-I
3
MMLE-II
MMLE-II
MMLE-II
*
QRE:
ME:
MME:
MMLE:
-lor -II :
r
r
t
1
2
3
-
quantile regression estimatio'n
method of moments estimation
modified method of moments estimation
modified maximum likelihood e:stimation
first or second modification type using rth order
statistic where r=1, 2, or, 3,.
a dash (-) signifies that the estimation algorithm converged
for fewer than 50% of the samples.
69
used to compile Table 4.6 were associated with the same data base except
those computed from the QRE's.
Thus, it should be understood that the
EIE's of Cohen and Whitten's 13 alternative methods can be compared
amongst each other with somewhat more control than is possible. in
comparing them with the EIE's of the QRE procedure.
It should also be
noted that in computing the EIE's of the QRE procedure, only results for
n=99 were used, since Cohen and Whitten's sample sizes were of this same
magnitude (n=100).
The estimates in Table 4.6 reveal that the efficiency of the QRE
procedure compares well in most instances to those of the thirteen other
methods.
Over the four 3PLN distributions the estimated index for the
QRE procedure ranked seventh,
ively.
~'"h.ile
first,
fourth and fifth-best, respect-
these comparisons would be more meaningful had the QRE
results pertained to Cohen and Whitten's actual data, they still offer
strong evidence that the efficiency of the QRE procedure is comparable
to those of many alternatives to the LMLE procedure.
70
4.2
Hill's (1963) Epidemic Data (3-Parameter Lognormal Distribution)
In Hill's (1963) paper, he showed that the conventional solution to
the likelihood equations of the 3-parameter lognormal distribution leads
to local rather than global maximwn
maximum likelihood estimates of
like~lihood
(Y,IJ,o)
estimates.
The global
were
shown to be (x ,-ao,+<Xl)
1
where xl is the first sample order statistic.
To avoid this unwanted
solution, Hill estimated the parameters u:sing Bayesian techniques.
The
data used by Hill have subsequently been used by other researchers in
applications of alternative estimation methods [Wingo (1975), Giesbrecht
and Kempthorne (1976)].
the
QRE procedure and
alternative
In this section, Hill's data will be applied to
the
results will be compared to those of the
methods used by others.
These alternative methods were
reviewed in Chapter 2.
Hill's
data
are
well-suited
because they are grouped.
for quantile regression estimation'
e
The data consist of 310 observations on the
incubation period (rounded off to the nealrest day) of inoculated smallpox virus.
The data are given in Table 4.7.
The value of Y, if the
distribution were actually 3-parameter logJlormal, was considered by Hill
to be -4, corresponding to the day of inoculation.
be somewhat greater than -4,
In reality, y should
since the onset of smallpox cC!uld not
possibly occur immediately upon inoculation.
Although the data are grouped, a more aggregated grouping at the
extremes seemed to be preferable for use in the QRE procedure.
sets
of
groupings
were
10, and 11 quantiles.
tried
corresponding to
These groupings
the
Four
choice of 8, 9,
are shown in Table 4.8.
Note
that in each case the last observation (X = 19 days) was dropped from
e
71
Table 4.7.
Day (X)
Hill's (1963) Data on Incubation Period of
Inoculated Smallpox
Observed
Frequency
Cumulative
Frequency
2
6
2
8
25
102
198
271
293
301
304
307
-4
-3
-2
-1
0
1
e
2
3
4
5
6
7
8
9
10
17
77
96
73
22
8
3
3
11
12
13
14
15
16
17
18
19
1
308
309
1
310
1
72
Table 4.8.
Additional Groupings of Hill's Data into Quantiles
Cumulati've Frequency
Day
eXpI..)
8
quantiles
1.5
10
quantiles
quantiles
2
2
2
9
quantiles
11
2.5
8
8
8
8
3.5
25
25
25
25
4.5
102
102
102
102
5.5
198
198
198
198
6.5
271
271
271
271
7.5
293
293
293
293
8.5
301
301
301
301
9.5
304
10.5
307
307
11.5
12.5
308
308
308
308
13.5*
309
309
309
309
* This
quantile corresponds to p.
discarded from the regression.I.
=1.0
and was therefore
73
the data.
e·
The assumption was that such a large outlier would introduce
unwanted distortion into the estimation process.
The results of applying the QRE procedure to each of the four
aggregated datasets are presented in the upper portion of Table 4.9.
This table gives the parameter estimates, and their asymptotic standard
2
errors as well as X2 values and the probability of greater X values
corresponding to a goodness-of-fit test for 3-parameter lognormality.
The lower portion of Table 4.9 presents the estimates and goodness-offit test results that apply to analyses of these data found elsewhere in
the literature .
. The algorithm described in Section 1 of Appendix
perform
the
quantile regression estimation.
(see Appendix B) were used as start values.
a
was used to,
Pseudo-moment estimates
The standard error esti-
mates were taken. as the square roots of the diagonal elements of the
estimated
asymptotic
covariance
matrix A
of
the
parameters.
The
matrix A was computed according to
(4.10)
where F is the estimated matrix of partial derivatives of the regression
equation with respect to the parameters, V is the estimated covariance
matrix of the quantiles,
and s2 is the variance of the regression
residuals.
2
The X value for the goodness-of-fit test was computed according to
" 2"
= I[(f.1
- f.) If.]
11
(4.11)
where k was the number of quantiles, f. was the observed frequency in
1
the ith grouping interval, and f. was the predicted frequency of the ith
1
74
"
Table 4.9. Summary Statistics for Various 3PLN Estimations of Hill Data.*
2
X
Estimation Method
y
QRE: 8 quantiles
-2.09
(4.11)
1.96
(.589)
.192
(.113)
18.10
.0012
9 quantiles
-4.02
(6.31)
2.20
(.706)
.154
(.109)
16.48
.0024
10 quantiles
-4.11
(6.10)
2.21
(.674)
.152
(.103)
16.41
.0025
11 quantiles
-3.79
(5.61)
2.18
(.645)
.159
(.103)
16.79
.0021
(j
Prob>x
2
.,
-------------------------------------------------------------------------
e
Other Methods (using 8 quantiles)
"True" from Hill
-4
2.20
.167
21.57
.0002
Bayesian LMLE from Hill
-2.41
2.01
.204
23.03
.0001
Discrete MLE from
Giesbrecht & Kempthorne
-1.65
1.90
.222
21. 75
.0002
LMLE via penalty function
from Wingo
-2.45
2.01
.203
22.76
.0001
.
2
*Standard Errors in parentheses; all x -values computed with respect to
8-quantile grouping.
75
e.
grouping interval according to the estimated parameters.
In all of the
X2 calculations of Table 4.9, k was set equal to eight, corresponding to
the observed frequencies in the 8-quantile grouping.
The most striking information to be learned from Table 4.9 is that
Hill t s data was almost certainly not lognormal!
All procedures gave
2
fairly similar parameter estimates and X values in the goodness-of-fit
tests.
2
The QRE procedure using 10 quantiles gave the smallest X value.
Ho~ever,
even this value rejected the hypothesis of 3-parameter lognor-
mality at the .01 level of significance.
It is unfortunate that several researchers have used this data to
apply 3PLN estimation procedures.
Still, the
data~
do appear to come
from a distribution similarly shaped to the 3-parameter lognormal, and
for these data the QRE procedure certainly compared favorably to the
procedures espoused by Hill
(Bayesian MLE), Wingo
(MLE with. penalty
function) and Giesbrecht and Kempthorne (discrete MLE).
2
In fact, the X
values corresponding to each set of QRE results were smaller than any of
those given by the various MLE procedures.
Thus, the QRE procedure
would seem to be quite useful as long as it performed this well on data
that were truly 3PLN-distributed.
76
4.3
Monte Carlo Study II--Six SB Distributions from Johnson and Kotz
(1970)
In this section, the sampling behavio:r of quantile regression es-
timates of the parameters of the SB distribution is examined in comparison with
that
The data
of local maximum likelihood estimates.
consisted of 100 independently generated raIldom samples of size 299 from
each of six SB distributions.
Both the QRE and LMLE procedures are most easily implemented conditionally on fixed values of the range
par~~eters,
a
and~.
To estimate
all four parameters, a grid search must be conducted in the a,
~
plane
to find that pair (a, 13) for which the gll:>bal minimum (or maximum) of
the appropriate objective function occurs.
LMLE
algorithms
are
given
in
The details of the QRE and
Section!; 3
and
4
of
Appendix B,
respectively.
The six SB distributions investigated in this small Monte Carlo
study were taken from Johnson and Kotz [1970, pp. 24-25].
Each distri-
bution is bounded below by zero and abov'e by one, but the six vary
considerably
in
skewness
and
kurtosis.
functions of each are plotted in Figure 4.3.
The
shape
of
the
density
The parameter values of IJ
and a are given at the top of each plot.
The samples of size 299 were created by generating vectors of 299
independent standard normal values, z, and then forming SB-distributed
v~lues,
x, via the transformation
x = (a + ~y)/(l +
y),
(4.12)
where
y
= exp(1J
+ O'z).
(4.13)
e
e
e
MU-O..
21,
SIGMA-2
,
,I
MU-O..
SIGMA-SORT(2)
21,
-
"'--"
"'--"
'+-1
'+-1
,
MU--1.066~
r--r'--~==r=====;~~
,I
H
x
SIGMA a 2
51
4 I
" I
H
'l;:'3
2
o"
o
,
x
o"
"
i
i
i
,
x
.
,
o"
MU--I, SIGMA-I'
MU-O.l SIGMA-.S
41
o
1
41
i
i
i
,
x
"
I
MU--.S .. SIGMA-.S
4
I
~3
'+-
H:3
1<3
"'--"
"'--"
2
2
"'--"
o
'+-
I i ,
,I
'+-
2
1
0'
''<
o
Figure 4.3.
,
x
>m'
,
0"
o.
':=t'
,
x
0'
'>:cm'
,,,<
o
,
x
Probability density functions of six S8 distributions.
.......
.......
78
The sample size of 299 was decided upon after preliminary exploration
revealed that the estimation algorithms frequently could not converge
for smaller sample sizes (n=99) from the fourth and sixth of the six
distributions.
In forming the quantiles, the followi.ng vector of quantile points,
p, was used:
p
= (.02,
.08, .16, .26, .40, .60, .74, .84, .92, .98).
As in the case of the 3-parameter lognormal distribution, p was chosen
somewhat arbitrarily, but with the intent of capturing information near
the extremes of the data.
In this case, however, p. was· chosen to' be-
SYmmetric about
three
.50,
since
of the six distributions were sym-
metric.
The means and standard deviations of the SB parameter estimates
from the QRE and LMLE procedures are sho\irn in Table 4.10.
This table
also shows the true parameter values of eac:h distribution and the number
of "runs" out of 100 for which the estimation algorithm converged.
In
this study, the estimation was halted if the estimate of a fell below -2
or if the estimate of
estimates
~
rose above 3.
would occasionally diverge
Without imposing these bounds the
toward plus or minus
infinity.
Inspection of Table 4.10 reveals thalt for most of the distributions, both estimation procedures were realsonably accurate and precise.
Clearly,
the QRE procedure was
somewhat less precise than the LMLE
procedure, but in these cases the inferi.)r precision did not lead to
less accurate mean estimates.
ficantly biased estimates of
Both procedures did appear to give signi-
a, however, for distributions 4 and 6.
In
each case the true value of a was 0.5 while the estimates of a centered
e
79
Table 4.10.
Means and Standard Deviations of SB Parameter Estimates
from LMLE and QRE Procedures
= LMLE
Method
Parameter
True
Value
Mean
Standard
Deviation
Method
Mean
= QRE
Standard
Deviation
Distribution 1
a
0
1
0
2
0.0014
1. 0061
-0.0404
1.9424
(Runs
.0028
.0029
.1096
.0591
= 100)
-0.0016
1. 0012
0.0041
2.0236
(Runs
a
0
f3
·1
0.0082
0.9921
0.0055
1.4631
(Runs
.0108
.0105
.0882
.0711
100)
=
0.0003
0.9973
0.0139
1. 4610
(Runs
.1380
.1606
= 100)
0
1
-1.066
2
-0.0009
0.9980
1.0351
1. 9614
(Runs
.0013
.0083
.1093
.0581
= 100)
-0.0021
1.0032
-1.0456
2.0051
(Runs
.0069
.0318
.1680
.1817
= 100)
0
1
0
.5
0.0254
0.9736
0.0007
0.5968
(Runs
.2388
.2377
.1656
.1455
= 98)
0.0018
0.9921
0.0080
0.5998
(Runs
.2338
.2213
.2186
.1927
= 85)
0
1
-1
1
0.0069
0.9552
-0.9527
1.0612
(Runs
.0109
.0731
.1167
.0869
= 100)
-0.0036
1.0331
-1.0046
1.0198
. (Runs
=
.0284
.1983
.1994
.1729
99)
0
1
-.5
.5
0.0368
0.9290
-0.4461
0.6063
(Runs
.0733
.2115
.2067
.1304
96)
0.0185
0.9881
-0.4697
0.6042
(Runs
.1200
.3389
.2811
.1800
= 83)
f3
lJ
a
Distribution 2
lJ
a
0
1.4142
Distibution 3
a
~
e
IJ
a
Distribution 4
a
~
lJ
a
Distribution 5
a
f3
lJ
a
Distribution 6
a
~
IJ
a
=
.0150
.0120
.1535
.1643
= 100)
.0326
~0301
.;
80
around 0.6.
For all other parameters, though, the mean estimates were
very close to the true parameter values.
The precision of both proce-
~
dures, especially that of the LMLE procedure, was remarkably good in the
first
three
kurtotic)
distributions,
distributions.
but declined for the latter three
These
subjective
(more
observations will now be
refined by a more detailed analysis of the results.
Table 4.11 provides further insight into evidence of bias from each
procedure.
This table lists, for each procedure and each distribution,
the mean parameter estimates, the value of the t-statistic corresponding
to a test for unbiasedness and the probability of obtaining a greater
absolute value of t.
As' expected, the p,rocedures were significantly
biased in the estimation of
0
in distributions 4 and 6.
But whereas
there was little evidence of significant bias among the remaining quantile regression estimates, there was very strong evidence of bias in the
majority of the remaining local maximum likelihood estimates.
degree of bias in the remaining LMLE' s was small,
Since the
this statistical
e
significance only indicates that the estimation procedure is precise
enough to detect a small bias for sample sizes of 299.
The QRE's did
not appear to be much more accurate than the LMLE's, but their variances
were
usually
too
large
to
render
thesE~
inaccuracies
statistically
significant.
Since the results
show no evidence that one procedure is more
accurate than the other, the precision of the QRE procedure relative to
the LMLE procedure would seem to serve as a valid criterion for evaluating the relative merits of the two procedures.
Estimates of the
relative precision associated with estimntes of each of the four SB
parameters are given in Table 4.12.
~
81
Table 4.11.
Tests for Bias in SB
Parameter Estimates from LMLE and QRE
Procedures (Sample Size = 299)
= LMLE
Method
Parameter
True
Value
Mean
Estimate
t
Prob>ltl
0
1
0
2
0.0014
1.0061
-0.0404
1.9424
5.13
20.78
-3.69
-9.75
<.0001
<.0001
.0002
<.0001
0
1
0
1.414
0.0082
0.9921
0.0055
1.4631
7.61
-7.58
0.62
6.88
-0.0009
0.9980
-1. 0351
1. 9614
0
1
0
.5
Method
Mean
Estimate
= QRE
t
Prob>1 t'
-0.0016
-1.0012
0.0041
2.0236
-1.07
0.99
0.27
1.44
.2831
.3216
.7870
.1510
<.0001
<.0001
.5355
<.0001
0.0003
0.9973
0.0139
1.4610
0.10
-0.89
1.01
2.91
.9203
.3755
.3149
.0036
-6.86
-2.42
2.83
-6.63
<.0001
.0151
.0047
<.0001
-0.0021
1.0032
-1.0456
2.0051
-3.01
1.01
1.21
0.28
.0026
.3145
.2245
.7798
0.0254
0.9736
0.0007
0.5968
1.05
-1.12
0.04
6.59
.2930
.2637
.9668
<.0001
0.0018
0.9921
0.0080
0.5998
0.07
-0.33
0.34
4.77
.9435
.7434
.7348
<.0001
0
1
-1
1
0.0069
0.9552
-0.9527
1.0612
6.30
-6.13
4.06
7.05
<.0001
<.0001
<.0001
<.0001
-0.0036
1.0331
-1.0046
1.0198
-1.25
1.66
-0.23
1.14
.2114
.0965
.8201
.2546
0
1
-.5
.5
0.0368
0.9290
-0.4461
0.6063
4.92
-3.29
2.56
7.99
<.0001
.0010
.0106
<.0001
0.0185
0.9881
-0.4697
0.6042
1.41
-0.32
0.98
5.27
.1598
.7490
.3260
<.0001
Distribution 1
ex
~
IJ
(J
Distribution 2
ex
~
IJ
(J
Distribution 3
e
ex
~
IJ
(J
0
1
-1.066
2
Distribution 4
ex
~
IJ
(J
Distribution 5
ex
~
IJ
(J
Distribution 6
ex
~
IJ
(J
82
Table 4.12. Estimated Relative Precision of QRE's to LMLE's for
Six SB Distributions*
* Estimated relative precision of QRE procedure to LMLE procedure with respect to estimates of a parameter e is defined as
R.P. (8)
=
s.d. (8)LMLE
x
s.d. (8)QRE
where s.d.
= standard
deviation.
83
The relative precision CR.P.) estimates in Table 4.12 vary considerably across the four parameters and down the six distributions.
patterns do exist, however.
First, down the distributions, the relative
precision varied most for a and
~,
and least
for~.
Second, for a given
distribution, the relative precision was usually lowest for a and
highest
for~.
Some
~
and
Finally, the precision of the QRE procedure was closer
to that of the LMLE procedure when the kurtosis parameter cr was small.
Since smaller values of cr also corresponded to poorer precision in both
procedures, there was a negative correlation between the precision' and
the relative precision of the quantile regression estimates.
For distributions that could be estimated more precisely Ci. e. ,
when cr was high) the QRE procedure had poor relative precision, sometimes as
low as 18 percent.
Yet, since this low relative precision
corresponded with high absolute precision, it may be said that the QRE
procedure
performed
ordinarily well.
well,
but
the LMLE procedure performed extra-
84
4.4
1960-1972 U.S. Family Income Data--Inlcome Distribution Proposed by
Singh and Maddala (1976)
The income distribution proposed by Singh and Maddala (1976) was
presented
in Section 4 of Chapter 3.
The distribution function was
given 'as
x>
0, a > 0,
Upon differentiation of the above function,
C
> 0.
(4.1)
the density function was
found to be
(4.2)
Finally the quantile regression equation
x . =
p1
where
a*
WclS
{a*[(l _p.)-c* - l]}b* +
=11a,
1
b*
= lIb,
and c*
derived as
(4.3)
101.,
1
= l/c.
e
In their 1976 paper, Singh and Maddala estimated the parameters of
this distribution by employing the DavidoIL-Fletcher-Powell algorithm in
a nonlinear regression program to minimize the quantity
k
I {In[l - F(x .)] +
p1
i=l
C
In(l + axbp1.)}2
(4.4)
This estimation procedure ignores the fact. that the X . are correlated
P1
and heteroskedastic.
Thus, the results given by this procedure are not
expected to be as efficient as those obtainable via the QRE procedure.
The purpose of the following analysis is 1:0 compare the performance of
the QRE procedure [and also its unweighted (ordinary QRE) counterpart]
with that of the procedure used by Singh and Maddala [EC!.. (4.4)].
e
85
The
Notice
income data used in this analysis are shown in Table 4.13.
that
as
nominal
incomes
rise,
the number of income classes
gradually increases from eleven in 1960-1967 to twelve in 1968-1970 to
thirteen in 1971 and finally to fourteen in 1972.
While the
dat~
were
taken from the sources referenced by Singh and Maddala, there is no
guarantee that the numbers given in Table 4.13 are identical to those
used by Singh and Maddala.
A telephone
conversation with Professor
Maddala revealed that it would be virtually impossible to recover the
exact data used in his article.
The first analysis performed on the data in Table 4.13 consisted of
an attempt
to
duplicate. Singh
and Maddala' s
results
by estimating·
Eq. (4.4) using Marquardt's algorithm in the NLIN procedure of SASe
In
the process of doing so, it was deduced that Singh and Maddala must have
scaled their data by diViding each income by the median income for that
year.
This deduction was verified by the finding that the parameter
est~ates
given in Singh and Maddala's paper predicted a median income
very near to 1.0 for each of the thirteen years.
The
results
given in Singh
of the estimation of Eq. (4.4) .together with those
and
Maddala's
paper
are
shown
in Table 4.14.
The
2
estimates for each year are accompanied by a X goodness-of-fit statis tic corresponding to the quantity
= I[(f.-f.)
J J
A
2
A
If.]
J
where f. and f. are the observed and predicted frequencies, respectJ
J
2
ively, of the jth income class. For all X calculations, frequencies
were predicted for the eleven income intervals given in Table 4.13 that
86
Table 4.13. 1960-1972 U.S. Family Income Data* (Table entries are
number of families in correspoll.ding income range.)
Ye:ar
Income Range
1960
1961
1962
1963
1964
1965
1966
1967
0- 999
1000- 1999
2000- 2999
3000- 3999
4000- 4999
5000- 5999
6000- 6999
7000- 7999
8000- 9999
10000-14999
>15000
2285
3613
3970
4456
4773
5839
4889
3973
5135
4795
1707
2316
3573
4037
4387
4845
5439
4714
4231
5375
5219
2205
1950
3469
3901
4325
4669
5424
5100
4023
5804
6019
2314
1791
3250
3792
4142
4287
5253
4844
4300
6335
6857
2585
1532
2988
3864
4001
4113
4738
4714
4458
6635
7761
3031
1459
2956
3583
3806
3883
4502
4477
4683
6952
8342
3636
1149
2635
3197
3341
3474
4108
4574
4542
7408
10008
4486
1031
2189
2981
3155
3243
3879
4145
4414
7661
11147
5989
45435
46431
46998
47436
47835
48279
48922
49834
5620
5737
5956
6249
6569
6957
7532
7933
Total
Median
Year
Year
Year
Income Range
1968
1969
1970
Income Range 1971
Income Range 1972
0- 999
1000- 1999
2000- 2999
3000- 3999
4000- 4999
5000- 5999
6000- 6999
7000- 7999
8000- 9999
10000-14999
15000-24999
>25000
909
1717
2756
3081
3031
3485
3839
4142
7678
12625
6112
1313
804
1600
2371
2705
2752
3033
3281
3726
7389
13682
8005
1889
829
1535
2237
2613
2728
3026
3122
3294
7054
13925
9193
2392
784
0- 999
1000- 1999 1339
2000- 2999 2242
3000- 3999 2574
4000- 4999 2888
5000- 5999 3027
6000- 6999 2955
7000- 7999 3327
8000- 9999 6560
10000-11999 6686
12000-14999 7674
15000-24999 10399
>25000
2841
Total
50510
51237
51948
683
0- 999
1000- 1999 1183
2000- 2999 2018
3000- 3999 2494
4000- 4999 2670
5000--5999 2735
6000- 6999 2828
7000- 7999 ·3030
8000- 9999 6063
10000-11999 6245
12000-14999 7947
15000-19999 8597
20000-24999 3899
>25000
3982
8632
9433
9867
Median
*
Total
53296
Median
10285
Total
54373
Median
10650 t
Source: U.S. Bureau of the Census Current Population Reports, Series
P-60 and P-20, 1960-1972.
t Projection, no data available.
e
e
87
Table 4.14. Parameter Estimates of Income Distribution of
Singh and Maddala Obtained from Singh and Maddala's
Estimation Procedure
Parameter Estimates
Year
1-
b
c
2t
X
Results Reported in Singh and Maddala's Article
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
2.
a
.2931
.2735
.3079
.3084
"~3184
.3082
.3109
.3120
.3071
.3101
.3102
.3125
.3070
'
1.992
1.972
2.063
2.051
2.080
2.127
2.197
2.012
2.111
2.131
2.121
2.139
2.064
2.803
3.009
2.609
2.597
" 2~550
2.624
2.558
2.552
2.712
2.611
2.546
2.544
2.538
1798
1620
1635
1557
1332'
1926
2167
1035
953
1225
1428
1448
846
2.459
2.364
2.942
3.846
5.059
5.027
8.862
10.204
2.181
2.576
2.997
3.719
3.894
2603
2280
1363
1036
565
511
572
414
1686
1241
875
514
397
Results Obtained in Present Estimation
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
.3385
.3578
.2760
.2037
.1515
.1568
.0848
.0701
.3928
.3228
.2715
.2134
.1863
2.130
2.081
2.023
1.961
1.899
1.899
1.882
1.872
2.261
2.144
2.033
1.946
1.927
tAll X2 values' are significant at the .01 percent level indicating
that the assumed functional form of the income distribution was at
best a useful approximation to the true distribution.
88
correspond to the years 1960-1967.
Theref,ore, there are seven degrees
2
of freedom associated with each X statisti.:.
Notice that for each year,
the results obtained here differ from
those reported by Singh and Maddala.
Also note that for nine of the
thirteen years, the estimates obtained here provided a better fit to the
data than did those given in Singh and Madldala' s paper.
that
either
presented
Singh
and
Maddala' s
Table 4.13,
in
or
that
data
dliffered
Singh
and
This suggests
somewhat
Maddala
from
that
modified their
estimation procedure somewhat, perhaps to enable themselves to obtain
estimates that remained relatively stable across years.
That the latter
may have been donei"fJ" 'suggested by the much smaller across-ye'ar variation in their parameter estimates relative to those obtained here.
computed
2
X
statistics
in
all
cases
were
very
The
highly significant,
indicating that U.S. family income is not distributed according to this
distribution.
However, as Singh and Maddala point out, this distribu-
tion does provide a better fit to the data than do either of the Pareto
or
lognormal distributions,
and thus
represents
an improvement over
these two conunonly assumed distributional f4:>rms.
Next, the QRE procedure [Eq. (4.3)] w.!s applied to the data using
both iteratively reweighted, and ordinary llonlinear least squares.
For
each year, the dependent variables in the regression, X ., consisted of
p1.
the
upper endpoints of the
malized
by
the
median
income classes listed in Table 4.13 nor-
income
variables, p., were computed as
1.
-1
p. = N
1.
i
I f.
j=l J
for
that.
year.
The
"independent"
e
89
where N was the total number of families in "the population and f. was
J
the number of families falling in the jth income class.
The number of
observations in each year's regression was one less than the number of
available income classes since no quantile point could be obtained for
the last class.
A Gauss-Newton algorithm simiiar to that documented in
Section 1 of Appendix B was used to perform the iteratively reweighted
nonlinear least-squares
2
and X
statistics
(IRNLS)
regressions.
The parameter estimates
from each group of regressions are presented in
Table 4.15.
For two of the years of data, 1966 and 1967, the IRNLS failed to
converge•
•
For these" two years,
diverging toward infinity.
estimation results
a "converged
toward zero while'"
c "was
Inspection of the data in Table 4.13 and the
in Table 4.15
reveals
that
there
is
a positive
correlation between the magnitude of
c and
lation in the highest income group.
Since this proportion was largest
the proportion of the popu-
in 1966 and 1967, it may be that the IRNLS algorithm failed for reasons
relating to the excessive size of this omitted group.
The correlation between the size of
c and
the portion of families
in the highest income class is disturbing because the class boundaries
were defined arbitrarily, and therefore should not be related to the
form
of
the
relationship
distribution.
is
that
there
A possible explanation for the apparent
might
exist paths
in the
(a,c)
plane
approaching the point (0, 0Cl) along which the shape of the distribution
changes very little.
Perhaps, when no quantile points are available
near the upper end of the distribution, the fit tends to be better at a
point further down one of the paths.
90
Table 4.15. Parameter Estimates of Income Distribution of Singh and
Maddala Obtained from Weighted and Ordinary QRE Procedures
Parameter Estimates
Year
1-
2t
X
~0253
L720
1.695
1.733
1. 714
1. 718
·1.704
12.155
9.382
9.187
16.072
20.033
29.377
1034
870
624
505
315
418
.1275
.1297
.1191
.1184
.0896
1.904
1.874
1.822
1.813
1. 788
5.983
5.877
6.360
6.416
7.724
393
451
372
299
184
b
Weighted QRE Procedure
1960
1961
1962
1963
1964
··1965
1966
1967
1968
1969
1970
1971
1972
2.
c
a
.0623
.0809
.0815
.0456
.0364
Ordinary (Unweighted) QRE Procedure
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
.1996
.2120
.1626
.0985
.0652
.0476
1.870
1.833
1.827
1. 780
1. 757
1.736
4.072
3.815
4.815
7.621
11.294
15.671
1252
1069
717
554
330
430
.2631
.1954
.1594
.1396
.1065
2.017
1.928
1.857
1.830
1.804
3.114
4.025
4.832
5.488
6.556
711
590
447
330
200
2
tAll X values are significant at the . 01 percent level indicating
that the assumed functional form of the: income distribution was at
best a useful approximation to the true distribution.
e
91
To summarize, the QRE procedure seems to work very well as long as
the objective of estimating this income distribution is to approximate
the shape of the distribution rather than to obtain precise estimates of
the individual parameters.
obtained,
the
2
X
In every year for which estimates could be
statistics
for
the QRE' s were smaller than those
obtained using Singh and Maddala's estimation procedure.
the
x.z
Furthermore,
values corresponding to the weighted QRE' s were always smaller
than those corresponding to the ordinary (unweighted) QRE's.
In future applications of the QRE procedure, it would seem important that the investigator try to obtain quantile points within the
extreme deciles of the distribution •. ' In the present analysis',
diffi~'
culties in obtaining estimates arose when 9.1 percent and 12.0 percent
of the population, respectively, fell in the final income group.
For
the eleven years in which estimates could be obtained, this uppermost
income group numbered less than 8 percent of the population.
92
4.5
Monte Carlo Study III--Application of Conditional QRE and Ordinary
QRE Procedures to the Four
3-ParameteI~
~
Lognormal Distributions of
Cohen and Whitten
In
(CQRE)
Section 3.5,
procedure
the
was
conditional
introduced
as
quantile
a
regression
possible
estimation
alternative
to
uncondi tional, or weighted QRE procedure ,:)utlined in Section 3.1.
the
The
usefulness of the CQRE procedure will depetld heavily on how the sampling
characteristics of its estimates compare "1ith those of the weighted QRE
procedure.
Also, since the CQRE is especially recommended when weighted
least squares
is
deemed undesirable,
it is important to compare the
properties of unweighted (or ordinary) CQRE' s with those of ordinary
quantile regression estimates (OQRE's).
The
Monte
Carlo
study
behavior of conditional QRE t
of
S
this
and
section
ordinaJ~
examines
QRE t s.
the
sampling
Each procedure was
applied
to the same 3PLN samples used in the Monte Carlo study of
Section
4.1.
99, 299,
Recall
499, 699,
that
and
899
the
data are
from
each
e
generated samples of size
of
four
3PLN
distributions.
Details of the data generation are given in Section 4.1.
The CQRE results are summarized in Tahle 4.16.
This table presents
the means and standard deviations of the CQRE parameter estimates and
the number of samples
verged.
(Runs) for which the estimation algorithm con-
In comparing these results with the LMLE and QRE results of
Tables 4.2 and 4.3 , it is found that the CQRE' s for these samples are
more biased and are estimated less precislely than either the LMLE' s or
the QRE t s.
For distributions 2 through 4 the bias of
y was
always
negative and ranged between three and tweln.ty percent of the true value
e
93
Table 4.16.
Means and Standard Deviations of 3PLN Parameter Estimates
Using Conditional QRE Procedure
Standard Deviation
Mean
Sample Size
n
y
Distribution 1
99
299
499
699
899
(y
= -10,
-9.482
-12.094
-11.269
-11.410
-12.858
~
= 2.2976,
2.009
2.291
2.298
2".323
2.439
= -3.1623,
Distribution 2
(y
99
299
499
699
899
-3.754
-3.459
-3.454
-3.309
-3.341
= -2,
Distribution 3
(y
99
299
499
699
899
-2.244
-2.102
-2.121
-2.057
-2.082
~
(y
99
299
499
699
899
-1.539
-1.469
-1.485
-1.446
-1. 462
= 1.1036,
.5701
.5965
.6220
.5975
.6206
.0710
.1365
.1762
.1540
.1813
(J
(J =
.6552
.5984
.4737
.4410
.4395
= .3087,
skewness
2.1376
1.1052
.9356
.6811
.4803
.5287
.3199
.2711
.2081
.1575
.4724, skewness
= .1438,
.6974
.6442
.6269
.6341
.6161
.9067
.4413
.3959
.2832
.2158
(J=
.0986
.0620
.0488
.0416"
.0359
76
87
88
94
99
= 0.980)
.1640
.0967
.0804
.0623
.0480
98
100
100
100
100
= 1.625)
.5027
.2782
.2343
.1739
.1382
.6368, skewness
.4397
.2443
.2245
.1589
.1246
Runs
= 0.301)
7.491
8.881
6.052
6.163
7.458
.5179
.4765
.4645
.4701
.4561
~
(J
~
.0998, skewness
.3472
.3110
.3032
.3072
.2967
= .5816,
= -1.4142,
Distribution 4
(J=
.1675
.1182
.1115
.1064
.0938
1.129
1.144
1.159
1.129
1.150
~
y
(J
~
.5783
.3111
.2546
.1841
.1513
.2158
.1240
~1029
.0780
.0614
100
100
100
100
100
= 2.475)
.2749
.1586
.1308
.0979
.0784
100
100
100
100
100
94
of y.
The precision of the parameter esti.mates for these three distri-
butions was usually only about half that l)f estimates given by the QRE
procedure.
a~nd
For Distribution 1, the bias
a~mong
estimates was not perceptibly different
procedures.
procedure
However,
converge
procedures
in only 76 samples:
when the
converged more
precision of the parameter
out of 100 could the CQRE
sample size was
often
(99
the CQRE, LMLE and QRE
and 83
99.
The LMLE and QRE
times,
respectively)
for
these samples.
The CQRE procedure might still be of s,ome value if it could prOVide
estimates
superior
to
those given by the OQRE procedure.
Ordinary
quantile regression estimation· would· be aIll obvious alternative whenever
the
weighted
important
to
QRE
procedure
determine
failed
which
to
estimates
converge.
are
Therefore,
second
best,
it
CQRE I
S
is
or
OQRE's.
The OQRE results are summarized in T,able 4.17.
2-4, the OQRE's for all three parameters
more precise than were the CQRE's.
Distribution 1,
but was
still
~l1ere
For Distributions
always more accurate and
This pattern was less pronounced for
somewhat
evident.
As an interesting
aside, comparisons of the OQRE's with the weighted QRE's of Tables 4.2
and
4.3
reveal strong evidence
for
the superiority of the weighted
results in both accuracy and precision.
Therefore,
the
results
of
this
MOD.te
Carlo study lead to
the
recommendation that the CQRE procedure not be applied to samples from
distributions similar to the four 3PLN distributions studied by Cohen
and Whitten.
considerations
If the
conditioning of
preclude
the
use
of
thE~
covariance matrix or other
wE~ighted
QRE,
then
ordinary
(unweighted) QRE should be attempted rather than conditional QRE.
95
Table 4.17.
Means and Standard Deviations of 3PLN Parameter Estimates
Using Ordinary QRE Procedure
Standard Deviation
Mean
Sample Size
n
Y
Distribution 1
99
299
499
699
899
(y
a
1.1
= -10,
-9.502
-12.490
-12.146
-11.286
-12.136
1.1
= 2.2976,
2.0238
2.3385
2.3614
2.3274
2.4101
= -3.1623,
Distribution 2
(y
99
299
499
699
899
-3.5039
-3.3977
-3.3363
-3.2616
-3.3010
Distribution 3
(y
99
299
499
699
899
-2.1127
-2.0880
-2.0763
-2.0371
-2.0660
Distribution 4
(y
99
299
499
699
899
-1.4623
-1.4542
-1.4593
-1.4347
-1.4538
Y
1.1
= -2, 1.1 = .5816,
.5780
.6056
.6118
.5938
.6142
.1266
.1611
.1733
.1546
.1762
1. 7132
.8915
.6521
.5519
.3891
= .1438,
.6842
.6360
.6300
.6346
.6197
.6767
.3615
.2862
.2233
.1716
skewness
.4188
.2656
.2021
.1695
.1296
.0802
.0564
.0441
.0367'
.0323
77
89
95
96
100
= 0.980)
.1281
.0788
.0619
.0503
.0386
99
100
100
100
100
= 1.625)
.3480
.2119
.1659
.1317
.1062
a= .6368, skewness
.3370
.1997
.1643
.1224
.0977
Runs
= 0.301)
.6161
.5787
.4828
.4066
.3834
a = .4724, skewness
.5110
.4714
.4677
.4710
.4593
1.1
7.877
8.668
7.408
5.535
5.771
= 1.1036, a = .3087,
.3409
.3075
.3060
.3080
.2994
1.1136
1.1442
1.1398
1.1211
1.1409
= -1.4142,
a= .0998, skewness
.1589
.1115
.1047
.1045
.0950
a
1.1
.3360
.2063
.1632
.1256
.1044
.1608
.0956
.. 0763
.0602
.0468
100
100
100
100
100
= 2.475)
.1966
.1165
.0938
.0728
.0572
100
100
100
100
100
96
CHAPTER 5
SUMMARY AND CONCLUSIONS .
The
quantile
regression estimation
veloped principally as
3-parameter
lognormal
3-parameter
log
a means of estimating the parameters of the
(3PLN)
and
related
equicorrelated -normal
Parameters
are
estimated
asymptotic
expected values
linear) least squares
(QRE) procedure has been de-
by
distributions
(3PLEN)
regressing
the
and SB distributions.
sample
quantiles
using' iteratively reweighted
(IRLS) techniques.
such as
Since
t~e
on
their
(often non-
data requirements
are minimal (say, on the order of ten sample quantiles), the procedure
is especially useful for preliminary analyses on large datasets that
have
yet
to
be
keypunched,
or for analyses
available in a condensed form.
of data that are
only
IRLS is necessary because the form of
the asymptotic covariance matrix of the regression residuals depends
upon the values of the unknown parameters.
The asymptotic means and variances of the quantiles can be obtained
as
long as the moments of the quantiles
moments of their
converge to the population
limiting multivariate normal distribution.'
For the
3PLN and 3PLEN distributions, the asymptotic expectations of the sample
quantiles were expressed in terms of the asymptotic moment generating
functions of corresponding standard normal quantiles, which posses the
necessary
directly
convergence
in
terms
of
properties.
The
quantiles
standard normal quantiles,
were
espressed
which also exhibit
moment convergence.
A more general application of the QRE procedure pertains to random
1
variables whose inverse distribution functions (F- ) are expressible in
closed form.
Provided moment convergence holds for quantiles from these
97
distributions, the QRE procedure applies straightforwardly since the
aSJDlPtotic expectations of the quantiles are directly obtainable from
F
-1
•
In a recent application, Koutrouvelis (1981) independently derived
the straightforward version of the QRE procedure for use in estimating
the 2-parameter Pareto distribution.
Ultimately, judgements of the value of the QRE procedure must be
based on the accuracy and precision of the estimates it produces.
In
Chapter 4 of this thesis, the sampling properties of QRE' s from four
3-parameter
compared
lognormal
to
those
of
and
six
estimates
estimation (LMLE) procedure.
SB
distributions
from
the
were
examined
local maximum
and
likelihood
In these Monte Carlo studies LMLE's were
obtained using whole samples, while of course, QRE's were derived using
selected sample quantiles.
•
The results showed that the QRE procedure performed well.
For
moderately and heavily skewed 3-parameter lognormal distributions using
sample sizes of 99 and greater, the estimated bias of the QRE' s was
small (less than 10 percent), and the relative precision of the QRE's to
the LMLE' s was never less than 60 percent.
Among the six SB distri-
butions, the QRE's showed little bias except in estimating the kurtosis
parameter a when kurtosis was fairly high.
The relative precision of
the QRE's to the LMLE's varied substantially among the six distributions
and four parameters, ranging from 18 to 106 percent.
Fortunately, the
low relative precisions corresponded to instances where the absolute
precision of the QRE's was very high.
Twice in this thesis, the QRE procedure was applied to real-world
grouped
data •
The 3-parameter lognormal distribution was estimated
using data from Hill (1963) and the income distribution of Singh and
98
Maddala (1976) was estimated using 1960-197-2 U.S. family income data.
For each of these applications
the estimates ·from the QRE procedure
provided a superior fit to the data according to the
did estimates
x2-criteria
e
than
from alternative estimation procedures reported in the
literature.
Amongst
the
various
applications,
the
iteratively
reweighted
nonlinear least squares algorithm converged most of the time.
In the
3PLN Monte Carlo studies, convergence was attained nearly always for
samples of size 299 or greater.
For samples of size 99, the algorithm
failed to converge for 17 of 100 samples from the least skewed of the
four distributions, but it converged in 299 of 300 samples from the
three more skewed distributions.
In the SIS Monte Carlo study, where all
sample sizes were 299, convergence was a slight problem for two of the
six distributions.
In a preliminary
anal~rsis
that used sample sizes of
99, the algorithm frequently failed to c1onverge.
It should be noted
that these two "problem" SB distributions were noticeably bell-shaped,
e
whereas the remaining four had shapes that. intersected the X-axis at a
more severe angle.
In the applications to real data,
co~~ergence
was attained for the
Hill data and for eleven of the thirteen years of U. S. family income
data.
The two years for which convergence failed had larger proportions
of families in the highest income category than did the other eleven
years, suggesting that the procedure performs best when the uppermost
quantile point is close to 1.0.
A conditional version of the QRE prclcedure was also developed in
this
thesis
as
an
alternative
lognormal distribution.
means
of
estimating
the 3-parameter
The conditional version has the advantage of
e
99
not requiring iteratively reweighted least' squares in the estimation
since much of the correlation among the quantiles is accounted for in
the
regression equation itself.
Unfortunately,
the results of this
procedure were quite poor in comparison with the unconditional QRE
results
and
even
in
comparison
with
unweighted
unconditional
QRE
results.
Finally, a Monte Carlo study was conducted to examine how quickly
the
moment
generating
functions
of
standard
normal
quantiles,
and
equivalently, the moments of 3PLN quantiles converge to their asymptotic
expected values.
While the rate of convergence depends on both the
quantile point, p, and the value of the_3PLN shape parameter,
it was
0,
found that on the whole, agreement. between the actual and asymptotic
moments of the quantiles was good for sample sizes of 59 or greater.
Convergence was found to be fastest when p was near 0.5 and when
° was
small.
In conclusion, the QRE procedure appears to be a valuable method of
estimation to use on quantile data from a large class of distributions.
The application is particularly attractive for the 3-parameter lognormal
and
SB
distributions,
for
which
the
regression
equations
may
be
expressed in terms of moment generating functions and expectations of
standard normal quantiles, respectively, each of which have well-behaved
asymptotic properties.
When ungrouped samples are available from these
distributions, local maximum likelihood estimation will tend to yield
more precise
estimates,
and
is
therefore
recommended
over
the QRE
procedure unless data reduction (selection of quantiles) is desired for
other reasons.
Further work on the determination of optimal quantile
spacings for condensing data might lead to significant improvements in
the precision of quantile regression estimates.
100
LIST OF REFERENCES
Aitchison, J. and J. A. C. Brown. 1957.
Cambridge University Press, London.
The
Lognormal
Distribution.
von Bertalanffy, L. 1938.
A quantitatbre theory of organic
(inquiries on growth laws, II). Humall Biology 10:181-213.
growth
and the Transformed
Blom, G. 1958. Statistical Estimates
Variable. John Wiley and Sons, Inc., New York.
Beta-
Bofinger, E. 1975.
Optimal condensation of distributions and optimal
spacing of order statistics. Journal of the American Statistical
Association 70(349):151-154.
Calitz, F. 1973. Maximum likelihood estimation of the parameters of
the three-parameter lognormal distribution--a reconsideration.
Australian Journal of Statistics 3:185-190.
Cheng, S. W. 1975. A unified approach to choosing optimum quantiles
for the ABLE's.
Jo.urnal of the American Statistical Association
70(349):155-159.
Cohen, A. C. 1951. Estimating parameters of logarithmic-normal distributions by maximum likelihood. Journal of the American Statistical
Association 46:206-212.
Cohen, A. C. and B. J. Whitten. 1980. Estimation in the three-parameter lognormal distribution. Journa,l of the American Statistical
Association 7~(370):399-404.
.
Cramer, H. 1946.
Mathematical Methods
University Press, Princeton, N. J.
David, H. A.
York.
1970.
Order Statistics.
of
Statistics.
e
Princeton
J()hn Wiley and Sons, Inc., New
Eubank, R. L. 1981. A density-quantile function approach to optimal
spacing selection. Annals of Mathema1t.ical Statistics 9(3) :494-500.
Finney, D. J. 1941. On the distribution of a variate whose logarithm
is normally distributed.
Supplement to the Journal of the Royal
Statistical Society VII:155-161.
Giesbrecht, F. and O. Kempthorne. 1976. 11aximum likelihood estimation
in the three-parameter lognormal distribution.
Journal of the
Royal Statistical Society, Series B, :38:257-264.
Griffiths, D. A. 1980. Interval estimation for the three-parameter
lognormal distribution via the likelihood function.
Journal o f _
Applied Statistics 29(1):58-68.
~
101
Harter, H. L. 1961. Expected
Biometrika 48:151-166.
values
of
"normal
order
statistics.
Harter, H. L., and A. H. Moore. 1966.
Local-maximum-likelihood estimation of the parameters of three-parameter lognormal populations
from complete and censored samples. Journal of the American Statistical Association 61:842-851.
Helwig, J. T., and K. A. Council -(ed.). 1979. SAS User's
Edtion. SAS Institute, Inc., Cary, North Carolina.
Guide--1979
Hill, B. M. 1963. The three-parameter lognormal distribution and
Bayesian analysis of a point-source epidemic.
Journal of the
American Statistical Association 58:72-84.
Iwai, S. 1950. Duration curves of logarithmic normal distribution type
and their application.
Memoirs of the. Faculty of Engineering,
Kyoto University 12:1-19.
Joluison, N. L. 1949. Frequency curves generated by methods
lation. Biometrika 36:149-176.
0'£
Johnson, N. L., and S. Kotz. 1970.
Continuous
tions-I. Houghton Mifflin, Boston.
Distribu-
Univariate
trans-
Kale, B. K. 1962. On the solution of the likelihood equations
iterative processes. Biometrika 49:479-486.
by
Kane, V. E. (forthcoming). Standard
and goodness-of-fit parameter
estimation methods for the three-parameter lognormal distribution.
Union Carbide Corporation, Nuclear Division, Oak Ridge, Tennessee.
Kendall, M. G. and A. Stuart. 1958.
Volume 1. Hafner, New York.
The Advanced Theory of Statistics,
Koutrouvelis, I. A. 1~81. Large sample quantile estimation' in Pareto
laws.
Communications in Statistical Theory and Methods A10(2):
189-201.
Lambert, J. A. 1964. Estimation of parameters in the three-parameter
lognormal distribution.
Australian Journal of Statistics 6(2):
29-32.
Lambert, J. A. 1970. Estimation of parameters in the four-parameter
lognormal distribution. Australian Journal of Statistics 12:33-43.
Michelini, C. 1972. The convergence pattern of the scoring method in
estimating the parameters of a lognormal function. Journal of the
American Statistical Association 67:319-323.
Mood, A. M., F. A. Graybill, and D. C. Boes. 1974. Introduction to the
Theory of Statistics. McGraw-Hill Book Company, New York.
102
Mosteller, F. 1946. On some useful 'inefficient' statistics.
of Mathematical Statistics 17:377-402 .
Annals
Munro, A. H., and R. A. J. Wixley. 1970. Estimation based on order
statistics of· small samples from a three-parameter lognormal
distribution.
Journal of the American Statistical Association
65:212-225.
Neter, J. and W. Wasserman.
Richard D. Irwin, Inc.
Ogawa, J.
1.
1974.
Applied Linear Statistical Models.
Homewood, Illinois.
1951. Contributions to the theory of systematic statistics,
Osaka Mathematical Journal 3 (2) : 175-213.
O'Neill, B., and W. T. Wells. 1972. Some recent results in lognormal
parameter estimation using grouped and ungrouped data. Journal of
the American Statistical Associateion 67(337):76-80.
Rao, C. R. 1952. Advanced Statis tical Ml!thods in Biometric Research.
John Wiley and Sons, Inc., New York.
Sarhan, A. E., and B. G. Greenberg.
1962.
Contributions
Statistics. John Wiley and Sons, Inc., New York.
to
Order
Sarkadi, K. 1974. On the convergence of the expectation of the sample
quantile, from Limit Theorems of JProbability Theory, P. Revez
(ed.). North-Holland Publishing Company, Amsterdam, 1975.
Selby, S. L. (ed.). 1971.
Standard Mathematical Tables--Nineteenth
Edition. The Chemical Rubber Company, Cleveland.
Tiku, M. L. 1968. Estimating the parameters of the lognormal distribution from censored samples. Journal of the American Statistical
Association 63:135-140.
U.S. Bureau of the Census. 1960-1972.
Current
Series P-60 and P-20, Washington, D.C.
Population,
Reports,
Walker, A. M. 1968. A note on the asymptotic distribution of sample
quantiles.
Journal of the Royal Statistical Society, Series B,
30:570:575.
Wilson, E. B., and J. Worcester. 1945.
The
normal
transform. Review of Economic Statist.ics 27: 17-22.
logarithmic
Wingo, D. R. 1975. The use of interior penalty functions to overcome
lognormal distribution parameter est~nation anomalies. Journal of
Statistical Computation and Simulation. 4: 49-61.
Yuan, P. T. 1933. On the logarithmic frequency distribution and the
semilogarithmic correlation surface.
Annals of Mathematical
Statistics 4:30-74.
103
APPENDIXES
104
APPENDIX A:
MAXIMUM LIKELIHOOD ESTIMATION IN THE 3-PARAMETER LOGNORMAL
AND SB DISTRIBUTIONS
The 3-parameter lognormal (3PLN) distribution with parameters y,
p,
and 0 and the SB distribution with parameters a,
~
and 0 are closely
related in that each may be readily transformed to the normal
distribution.
N(~,
Indeed,
and
X
has
Therefore,
N(IJ,
X has
the
the
SB
(see
(~,
0
2
)
distribution if In(X - y) ,..
distribution
if
In[(X - a)/(p - X)] ,..
it comes as no surprise to find that the same
peculiari.ties that Hill -(1963)
estimation
3PLN
~
Chapter 2)
discovered· in 3PLN· maximum likelihood
also
arise
in
SB
maximum
likelihood
estimation.
In this
parameter
appendix Hill's proof that there exists a path in the
space
sample xl' .. ", x
along
n
which the
like1ihclod' function of any orde'red
from the 3PLN distribution approaches +c» will be
presented followed by an analogous proof for the SB distribution.
the second proof it will be shown that
space
a
~
along
xl' or
which
p~
x "
n
the
likelihood
pat~hs
functi.:m
In
exist in the SB parameter
approaches
+c»
,as either
Fortunately, unless either skewness is near zero or
the sample size is very small, the likelihoc)d functions from 3PLN and SB
samples usually have just one local maximum.
The parameter estimates
corresponding to this local maximum are called local maximum likelihood
estimates.
suggest,
As the results of estimation ilCl Section 4.1 of this thesis
there is
no
reason to treat
thE~se
local maximum likelihood
estimates any differently than maximum likelihood estimates are normally
treated.
e
105
Maximum Likelihood Estimation in the 3-Parameter Lognormal Distribution
2
Let In(X - y) ~ N(~, cr).
The likelihood function of n independent
observations from this distribution is
, 2
2
L(xI, ... ,xn;Y'~'cr ) = L(y,~,cr )
= (2ncr2 )-n/2n (x. _ y)-lexp-(1/2cr2 )I[ln(x. - y) - ~]2. (A.I)
1
1
Defining
....
~(y)
= n
-1
(A.2)
Iln(x.-y)
1
and
(A.3)
it is easily seen that
(A.4)
It will now be shown that
lim
y+x
L**(y)
= ~.
(A.5)
1
Proof:
= n-II[ln(x.
_ y) _ p(y)]2
(A.6)
1
< n- 1Iln2 (x. - y)
-
~
1
2
In (xl - y), for y sufficiently near xl.
Hence,
L**(Y) = [a(y)]-n n (x.1 _ y)-l
> Iln(x 1 - y)l-nn (Xi - y)
(A.7)
-1
106
for y near xl'
Let xl - y
= e(y).
L**(y) ~ /lne(y)
Then
I -ne -1 (-;OK(y)',
(A.B)
e
where
n
K(y)
=n
(x. - y)
i=2
The right hand side of Eq.
zero.
-1
1
.
(A. B) approaches infinity as e approaches
Indeed, ignoring the constant term
Ilne(y)I = -lne(y)
~~(y)
and noting that
for 0 < e(y) < 1,
(A.9)
permits the limit to be written as
/lne(y) / -ne-l(y) = lim[f(x)/g(x)],
lim
e(y)~O
(A.lO)
x+0
where x = e(y), f(x) = x-I, and g(x) = [-lDl(X)]n.
L'Hospital's Rule may
be used to evaluate the limit in Eq. (A.lO):
IUD [f(x)/g(x)] = lim [~(x)/gn(x)]
x~O
(A.ll)
x~O
= lim
[l/(nlx)] = +
~,
where fn(x) and gn(x) refer to the nth derivatives of f(x) and g(x),
respectively.
By applying the result in Eq. (A.ll) to Eq. (A.B) it is clear that
L**(y) approaches
+co
as y approaches xl'
that if y = xl' then P(y) =
-~
and o(y) =
Note from Eqs. (A.2) and (A.3)
+~.
tit
107
Maximum Likelihood Estimation in the SB Distribution
Let In[(X - a)/(~ - X)]
.-J
N(IJ, 0'2).
The density function of X is
then
fX(x)
= (2ncr2)-\(~
- a)[(x - a)(~ - x)]-l
2
x exp-(1/2O' )ln{[(x -
(A.12)
a)/(~ - x)] - 1J}2,
and for n independent observations ordered as Xl' ... , xn the likelihood
function is
L(xl, ... ,x n ; a,~,IJ,cr2)
= L(a,~,IJ,cr2)
= (2na2 )-n/ 2(p
X
- a)Dn[(x. - a)(p
1.
~ x.)]-I (A.I3)
1.
2
exp-(1/2cr ) I In{[(x.1. - a)/(p - x.)]
- 1J}2.
1.
Defining
(A.14)
and
&2 (a,p)
=n-1
I {In[(x. - a)/(p - x.)] - p(a,p)}
1.
1.
2
(A. IS)
it is seen that
(A. IS)
-1
«[a(a,p)] -n IT [(x.1. - a)(p - x.)]
.
1.
A
It will now be shown that
lim L**(a,p)
~xl
= +00
(A.16)
108
and
L**(a,~)
lim
~-.x
=
~.
(A. 17)
n
Proof:
Case 1:
a-'x 1
= n
-1
- x.)] - p(a,~)}2
a)/(~
I{ln[(x.1. -
1.
(A.18)
2
< n- l Iln
[(x.1. - y)/(~ - x.) ]
.
-
1.
2
~ In [(xl - a)/(~ - xl)]
for a sufficiently close to xl'
for a near xl'
I In(x l
Let xl - a
- a)
HencE~,
= [&(a,~)] -n n
~
= In2 (xl
o
[(x. - a)(~ - x.)]
1.
I -n n [(xi -
- a)
= e(a).
-1
(A.19)
1.
a)(~ -
xi)]
-1
TIlen
where
K(a)
= nn
i=2
[(x. -
a)(~ -
x.)]
1.
-1
.
1.
Clearly the limit of the right-hand side o,f Eq. (A.20) as a-'x
is equil
valent to the limit of the right-hand side of Eq. (A.8) as Y-'X : Since
l
this limit is
p(a,~)
=
-~
~,
&(a,~)
and
Case 2:
&2 (a,~)
Eq. (A.16) has been
~-.x
=~
X
n
<
~
<
Note that if a
~.
n
= n -1
< n-
l
I {In[(x.1. -
a)/(~
- x.)]
l.
p(a,~)}
2
I In [(x. - Y)/(~ - x.)]
< In2 [(x
-
for all
prove~n.
l.
1.
n
- a)/(~ -
X )] ;
n
In2 (p
- xn )
2
= xl'
then
109
for
~
sufficiently close to x.
n
L**(a,~)
A
= [a(a,p)]
I
In(~
>
-
for p near x.
n
-n
n [(x.1
- xn )
I
Let p - x
-n
Hence,
- a)(p - x.)]
-1
(A.22)
1
-1
n [(x.1 - a)(p - x.)]
1
n = eCP).
Then
(A.23)
where
=
n-1
n [(x. -
i=l
a)(~
1
- x.)]
'.
-1
.
1
Again, the limit on the right hand side of Eq. CA.23) as
P~x2
is clearly
equivalent to the limit on the right hand side of Eq. (A.S) which was
shown to be
p
=xn ,
Therefore Eq.
+cD.
then p(a,p) =
If both
approaches
a~xI
+cD.
and
+cD
(A.17) has been proven.
and o(a,p) =
~~xn
Note that if
for all -~ < a < xl'
+cD
simultaneously, the likelihood function also
However, in this case lim
~x1
p~x
n
p(a,p) is undefined.
110
APPENDIX B:
SELECTED ESTIMATION ALGORITHMS
Each of the analyses presented in Chapter 4 required substantial
software development.
All of the analysis programs were written in SAS
(Statistical Analysis System) and executed on an IBM 3033 computer.
In
addition, several control programs written in SUPERWYLBUR were utilized.
Documented listings of the source codes of these programs are available
from
the
author
upon
In
request.
this
appendix,
representative
algorithms used in the iteratively reweighted least squares and local
maximum likelihood estimations in Chapter L. are· given.
B.l
Algorithm to Find Quantile Regressicm Estimates from 3PLN Samples
Given that a vector of k quantiles x
obtained,
the
following
steps
were
p1
'
followed
regression estimates of the parameters.
x
p2
... , x
'
in
pk
obtaining
has been
quantile
The notation used matches that
of Chapter 3.
1.
Compute
c.=$
1.
-1
the
"independent
variables"
for
the
regression--
(p.), k .. =[p.(l-p.)]/[n~(c.)~(c.)],
1.
1.J
1.
J
1.
J
for i,j=I,2, ... ,k, i<j.
2.
Compute pseudo-moment estimates E~stimates
e*
= (yk, ~, 0*) of
e = (y, 1.1, a) for use as start v8,lues in the nonlinear'estimation (see below).
3.
Set m = 0,
em = e*,
Vm =
identit~'
matrix.
A
4.
Set m=m+l.
based on
em- 1
Compute
em
and V 1.
m-
= I-step Gauss-Newton estimate of
Compute V using
m
em.
e
III
5.
If
Ym
is less than user-specified minimum permissible esti-
mate, then stop iteration and note that estimation has failed
7.
Output 8
m
as estimate of 8.
The Gauss-Newton procedure requires that starting values be supplied for the parameter estimates.
Also, in this generalized nonlinear
least squares situation, start values are required for the elements of
the variance-covariance matrix V of the quantiles.
Regarding the choice of start values for the parameters, it is best
if little work is
required to obtain them, as long as they are close
enough to the solution for the estimation algorithm to converge.
The
method of moments provides a relatively easy means of obtaining unbiased
(but inefficient) parameter estimates which might serve as start values.
Unfortunately,
obtainable
from
in
the
grouped
strictest
data
sense,
because
computation of the sample moments.
moment
the
estimates
grouping
are
not
precludes
the
However, a pseudo-moment estimation
procedure was developed whose estimates worked extremely well as start
values.
This pseudo-moment estimation procedure
is
so
named because a
pseudo-dataset of n observations is created from the quantiles and then
the
method
of moments is applied to the pseudo-data.
To form the
pseudo-data from a set of k quantiles, it is assumed that all of the
observations in the intervals between each pair of consecutive quantiles
fell exactly at the midpoints of these intervals.
The observations less
than the first quantile were assumed to equal an amount 50 percent less
than the first quantile.
112
To form the initial estimate of the variance-covariance matrix V,
This too, 'worked extremely well.
the identity matrix was used.
Since
it worked well, there did not seem to be any point in using a more
complicated expression.
A further justification for using the ideIitity
matrix is that it gives unbiased parameter estimates.
B.2 Algorithm to Find Local Maximum Likeli.hood Estimates from 3PLN
Samples
Given an estimate of y,
mates of
the conditi4:mal maximum likelihood esti-
~ and a2 are simply
P(y)
=
n
(l/n) l InCx.-y) and
i=l
~
CB.l)
n
o2 CY ) = (l/n) l [In(x.-y) _ p(y) ]2:.
i=l
~
Traditionally,
the
method
(B.2)
of scoring ha:s been successfully used to
compute maximum likelihood estimates from lognormal samples [see Lambert
(1964), Harter and Moore (1966), and O'NEdll and Wells (1972)].
the local maximum likelihood estimate of 6=:(y,
~,
Here
a) is found by search-
ing over the permissible range of y for that value yk for which the
triplet [yk,PCyk),o()'*)] maximizes the value of the likelihood function.
This method was easy to program and worked quite well.
The likelihood function can be shown to attain a local maximum for
that value of y for which {In[&(y)] + P(y)} attains' a local minimum.
The logarithm of the likelihood function of a sample from the 3-parameter lognormal distribution is
113
2
2 n
2
n
-(n/2)ln(2na ) - (1/2a ) I [In(x.-y)-~] - I In(x.-y).
i=l
~
i=l
~
(B.3)
When the likelihood equation is evaluated at the point [y, P(y), &(y)]
it simplifies to
(B.4)
This expression can be rearranged to
L[x l ,··. ,x n ; y, ~(y), a(y)] = -(n/2) [In(2n)+1] - n[ln&(y)+p(y)].
(B.5)
= Kl - n[lno(y)+p(y)],
where the constant K is independent of the parameter estimates.
l
the
likelihood function attains
a
Thus
local maximum when [In&Cy)+PCa)]
attains a minimum over a restricted range of y-values.
CAs y is allowed
to get very close to the first order statistics xl' the value of the
likelihood function corresponding to y tends to infinity).
In the algorithm that follows, the maximum of the likelihood function over the range of y-values in the interval (-13, x -.l0 Xl ) is
l
found and the triplet [Y*,PCy*),oCY*)] corresponding to this maximum is
considered to be the local maximum likelihood estimate.
= min(x i ;
1.
Compute
2.
Set initial limits within which the search for Y* will take
Xl
i=1,2, •.. ,n).
place:
yL = x 1 -
41 x li
YU = x 1 - . 10 1XII
Set r=l.
(lower limit).
CB.6)
(upper limit).
(B.7)
114
3.
Set D = (YU-yL)/lO.
Compute L*[P(Y), aCyl] corresponding to
Y = Y , yL+D, Y +2D, ... , Y ; where P(Y) and a(y) are defined
L
U
L
by Eq. (B.I) and (B.2) and where
(B.B)
L*[P(y), a(y)] = Ina(y) + P(y).
Define
(B.9)
y = Min{L*[p(y),a(y)]}.
If)'L <
4.
y=
Y<
yU then go to step 5..
Set
Y=
YL then set r=2r. If
yU then set r = (1/2)r;
Set YL= xl - 4r
5.
If
aL
lXli·
Return to step 3.
= y-D and yU = y+D.
If (YU-yL».OOI then return to
step 3.
6.
Take local maximum likelihood estimates to be y, P(y), a(y).
Of course the barrier of Eq. (B.7) call be adjusted as the user sees
fit.
For the parent distributions used in the Monte Carlo study of
Chapter 4, the barrier given in Eq. (B.7) worked well.
maximum was found less than this barrier value.
In all cases a
Also the initial esta-
blishment of a lower bound for y can be adjusted.
This initial bound
will not affect the final estimate of y, but only the speed at which the
estimate is found.
For small sample sizes: the likelihood surface tends
to be more irregular and several local maxima may exist.
In such cases,
the global maximum within the permissible range of y may be more easily
found if the stepsize D is decreased.
115
B.3
Algorithm to Find Quantile Regression Estimates from SB Samples
This algorithm implements a combined grid search and least squares
estimatio~
technique.
Of the four SB parameters (a,
p,
~
and a),
~
and
a are estimated via generalized least squares conditional on values of a
and
p.
A grid search is conducted in the (a,
of values a,
P) plane for that quartet
p, pea, P), cr(a, P) which yields predicted quantiles that
minimize the weighted sum of squared residuals from the observed quantiles.
The regression ·equation is given in Eq. (3.34) of Chaper 3.
estimation is iterative in two ways.
The
First, the region of the (a,
P)
plane within which the grid search is conducted must be successively
narrowed.
Second,
the estimated covariance matrix of the quantiles,
being a function of a, must be successively updated by the most recently
obtained estimate of a.
The following algorithm presumes a vector of k quantiles x
p1
' ... ,
x
from a random sample of size n from an SB distribution.
pk
1.
Define:
x
= (xp1
x
X
p2
pk
p = (PI P2
),.
Pk)'·
= 4J -1 (p).
Z = [1 I. z], where 1 is a k x 1 vector o~ ones.
K = k by k asymptotic covariance matrix of z with
z
typical element
k .. = [p.(I-p.)]/[n~(z.)~(z.)] for p. < p.
1J
1
J
1
J
where n is the sample size and
1
~
normal probability density function.
J
is the standard
116
Set the initial value of V,
the covariance matrix of the
quantiles, equal to the k-dimensional identity matrix.
2.
Define
the minimum and maximum allowable .estimates of the
range parameters, a and p.
This permits the algorithm to be
stopped whenever estimates of
'
X
pl
x
pk .
or whenever estimates of
are diverging toward
CI
are diverging toward
~t
For the six SB distributions (all with a = 0,
-OIl
or
+ao
or
P=
1)
estimated in Chapter 4, the bounds on the estimates were set
as follows.
'"
Lower bound on a:
'"
Upper bound on a:
'"
Lower bound on p:
at = -2,
an = xpI
- . Oi/Xpit '
p* = x
pk + .Ollxpki,
L
"'-
Upper bound on p:
3.
~= 3.
Set initial lower and upper limits of the grid in the (a,
plane.
P)
The grid search will begin within these limits, but
the limits will be allowed to change if they are later found
not to enclose the least squares solution.
For the six SB
distributions investigated in Chapter 4, the initial limits of
the grid were
a-dimension
Initial lower limit:
Initial upper limit:
an - .l(RANGE)
aU = an
a
L
=
p-dimension
Initial lower limit:
Initial upper limit:
where RANGE
= ptL Afi·
P:L = Pi
PlJ = Pi
+ . l(RANGE) ,
e
117
Also, choose the number of gridsteps, S.
The size of each
grid step is, in the a-dimension
and in the p-dimension
The
algorithm cannot work with S < 3.
however, that S be at least 5.
It is recommended,
For the estimations of Chapter
4, S = 7 was used.
4.
For
each
grid
point
~)
(a,
form the dependent regression
variable
y = In[(x
a =
and compute
- a)/(~ - x)]
[~(a, ~), a(a, ~)] where
Then compute
yea,
~) = ~(a, ~) + a(a, ~)z
= za,
and
Finally, compute
".""
I¥
,,#JIIJJ
-
-1
SSE(a, P) = [x - x(a, P)]' V
A""'"
#'IIttI
[x - x(a, P)].
118
5.
Let (a*, p*) equal the grid point
was a minimum.
(a,
13)
for which SSE
(a,
13)
If (a*, P*) is an interior point on the grid
then go to step 6.
If (a*, P*) is on the border of the grid
then go to step 7.
6.
.
If da + dP < .001 then go to step 9.
Otherwise recompute V
and narrow grid as follows:
a L = a* - da,
au
= a*
+ da,
and
PL = P* -
dp,
= p* +
dp.
Pu
Return to step 4.
7.
If a* <
cmU
at
then stop.
Note that estimation failed.
a*< .00001 then stop.
Note that estimation fa.iled.
If p* - pt< .00001 then stop.
Note that estimation failed.
If
If p* <
~
then stop.
Note that estimation failed.
8.
= a L then a L = otL - . 2 (RANGE) .
If a* = aU then aU = Min(a* + da, an).
If p* = PL then PL = Max(p* - dp, ~).
If p* = Pu then Pu = Pu + .2 (RANGE).
V = [0'2 (a*, P*)]K. Return to step 4.
9.
Take estimates of SB parameters to be
If a*
= x:k
Otherwise aU = a*
Otherwise PL = p*
Otherwise f3u = p*
Otherwise
a = a*, P = p*, p = ~(a*, P), a = O'(a*,
B.4
ot
L
- da.
+ da.
- dp.
+ dp.
p*).
Algorithm to Find Local Maximum Likelihood Estimates from SB Samples
The likelihood function of a sample xl' ... , x
bution is
n
from an SB distri-
119
L(x; a, p,
~,
-n/2
2
(2na )
a)
=
(B.10)
~
2
exp{-(1/2a )
[In
x. - a
p~- xi
(P - a)·
(x. - a)(p - x.) .
~
When a and p are known, the random variable y
N(IJ,
a2 ).
Therefore, conditional on a and
estimates of
~
= In[ eX
~,
~
- a)! (P
-
X)] ....
the maximum likelihood
and a are
pea,
~)
= n
-1
L y.
.
~
~
= n- 1 L. In
[(x. - a)/(~ - x.)]
~
~
~
(B.ll)
and
a(a, ~)
= n -1 L [Yo~ - pea,
~)]
2
(B.12)
i
=
n
-1
2
L {In[(x. - a)/(~ - x.)] - pea, ~)} .
~
i
~
In logarithmic form the conditional likelihood function may be written
(B. 13)
In L(a,
P) = -(n/2)[ln(2na2 )
+ 1] + n
In(~-a) - L. In[(x.-a)(p-x.)].
~
~
~
The local maximum likelihood parameter estimates may be found by
maximizing Eq. (B.13) via a grid search in the (a,
~)
plane.
The algo-
rithm follows the QRE algorithm fairly closely, except that the conditional objective functions differ.
Thus, the algorithm will be given
in terms of the QRE algorithm to save time.
Let xl' .... , x
n
be a random sample from an SB distribution.
1.
Perform step 2 of QRE algorithm.
2.
Perform step 3 of QRE algorithm.
3.
For each grid point (a,'~) compute
120
IJ = n-
1
~ [In(x i - a)/(l3 - xi)]
~
and
Evaluate Eq.
4.
(B.13)
using
ei,
13,
Let (a*, P*) equal the grid point (a,
was a maximum.
If (a*,
then go to step S.
~)
and
IJ,
13)
0:2
to obtain In
for which In Lea,
13)
is em interior point on the grid
If (a*, p*) is on the border of the grid
then go to step 6.
5.
If da +
dP <
.001 then go to step 7.
Otherwise narrow grid as
follows:
aL
= a*
aU
= a* +
- dei,
and
dei,
PL
= p*
~U
= p* + dp.
- dp,
Return to step 3.
6.
Perform step 7 of QRE algorithm.
7.
Take estimates of SB parameters t,o be
Return to step 3.
a = a* , P = p*, IJ = 1J(C1I\-, P*), & = a(a*, p*).
A
A
•
121
APPENDIX C:
SOME INITIAL WORK ON DETERMINING THE OPTIMAL SPACING OF
3-PARAMETER LOGNORMAL QUANTlLES
Whenever data are available in ungrouped, or raw form, use of the
quantile
regression
estimation
procedure
selection of quantiles from the raw data.
must
be
preceded
by
the
Immediately, the question
arises of how many quantiles to select, and which ones?
Recall that
choosing the largest possible set. of quantiles, i. e., the full set of
order statistic~, invalidates the QRE procedure.!
Thus, the number of
quantiles should be small relative .to the sample size.
Once the number
of quantiles k has been decided, there should be at least one spacing of
the k quantile points for which the information retained from the sample
about a particular parameter or parameters is a maximum.
If the QRE
procedure were applied to such optimally spaced quantiles, the precision
of the parameter estimates would be optimized as well.
In the
following discussion,
a spacing Sk is defined as Sk
{Pi} = PI' P2' ... , Pk where 0 < PI < P2 < ••• < Pk < 1.
=
When there is
no chance of ambiguity the k-subscript will be dropped to give a cleaner
notation.
Oga~a
(1951) has outlined the procedure for determining an optimal
spacing of k quantiles for an arbitrary continuous random variable X
with density g(x) and unknown parameter 8.
He defined the information
about 8 in a sample as 1(8), and in a set of quantiles as I (8), where
S
the quantiles have been formed according to the spacing Sk.
Ogawa
!The QRE procedure relies on the asymptotic multivariate normal distribution of the quantiles. Order statistics can not be used in the QRE
procedure because their joint distribution does not tend towards a
multivariate normal.
•
122
defined the optimal quantile spacing for
which
the
maximized.
relative
information
of
~l
the:
given k as that spacing for
quantiles,
was
Equivalently, the optimal spaci.ng is that Sk which maximizes
In order to compute IS (8)
for a given spacing of quantiles, the
joint density function of the quantiles must be computed.
While the
exact joint distribution of the quantiles i.s not known, their asymptotic
distribution,
assuming
g(x)
is absolutely continuous,
multivariate normal (see Theorem 3.2).
is known to be
Specifically, the joint density
function of the k quantiles for a given spacing Sk is
h(Xp1 " " ' xpk ) = (2n
-k/2
g1 g 2 ... g k[Pl(P2-P1) ... (Pk-Pk-l)(l-Pk)}
k
x exp{-!!
2
~
2
[I
-p·)(p·-p·
1)
i=l (p·+
1 1
1
1
1--
g.(x .- c.)
1
p1
-~
n
k/2
2
1
- 2
where,
as in Chapter 3,
formation in
{Xp1.}S'
c
i
is the Pith population quantile.
The in-
the set of quantiles determined by the spacing S,
about a parameter 6 is then defined, in thE! Fisher sense as
In general, however, finding I (6) is very difficult and often will
S
depend upon the value of the unknown parameter 6.
taneous
found
solution to the k equations aIS(IB)/api
and these
equations
lognormal distribution.
have
=0
Then,
the simul-
i=l, ... ,k must be
not been derived for the 3-parameter
Ogawa has derived the system of equations for
the normal distribution when either
~,
or Cf, or both are unknown.
Cheng
4It
.
.'
123
•
(1975) developed a procedure to follow for obtaining the system of
= a- 1f[(x-IJ)/a].
equations for any density function of the form g(x)
However, the 3-parameter lognormal distribution does not fit into this
class of distributions.
Thus far,
the only contribution that has been made towards the
determination of optimal spacings of 3PLN quantiles was the empirical
study of O'Neill and Wells (1972).
each
of
the
according to
results
three
parameters
two
types
of
indicated
that
the
They estimated the information about
contained
spacings,
in grouped
equal
asymptotic
samples
formed
and logarithmic.
Their
variances
of. the
parameter
estimates were smaller when the data were grouped in logarithmic rather
than equal intervals.
These results were obtained from samples having
considerable
(0 > .65).
skewness
The
advantage
of
the
logarthmic
grouping might not be as great for less-skewed distributions.
A small empirical study was conducted here to help indicate a
suitable value for PI'
location parameter
Whitten
(see
the first quantile point,
y for
Section 4.1).
the
in estimating the
four 3PLN distributions of Cohen and
The
study
consisted
of
selecting
quantiles from samples of size 299 from these four distributions.
10
For
each sample, five sets of quantiles were formed corresponding to five
spacings
that differed only in the first quantile point.
The first
quantile point PI was varied from 0.01 to 0.05 in increments of 0.01,
while the remaining quantile points were held fixed at the values given
in Section 4.1.
Ten samples from each distribution were generated and
the means and standard deviations of the QRE's corresponding to each of
the five spacings were computed.
that when PI
= O. 01,
The results, given in Table C.1, show
the parameters were estimated with the greatest
precision, and usually with the greatest accuracy.
124
•
Table C.1. Means and Standard Deviations of 3PLN Parameter Estimates
for Various Definitions of the First Sample Quantile (Pl)*
Standard Deviation
Mean
PI
y
Distribution 1
.01
.02
.03
.04
.05
(y
a
fJ
= -10,
-12.183
-10.481
-11.800
-14.872
-13.308
fJ
Y
= 2.2976,
2.337
2.260
2.370
2.445
2.426
a= .0998, skewness
.1061
.1100
.0996
.1000
.0989
= -3.1623, = 1.1036,
Distribution 2
(y
.01
.02
.03
.04
.05
-3.129
-3.233
-3.342
-3.317
-3.345
Distribution 3
(y
.01
.02
.03
.04
.05
-1. 976
Distribution 4
(y
.01
.02
.03
.04
.05
-1. 400
-1.415
-1.436
-1.430
-1.436
~
= -2,
-2.011
-2.053
-2.043
-2.054
1.083
1.112
1.148
1.140
1.148
~
.5605
.4363
.4542
.6887
.5852
.1520
.1908
.1858
.1905
.1926
a = .4724, skewness
.4756
.4685
.4584
.4610
.4590
= -1.4142, fJ = .1438,
.1305
.1437
.1633
.1584
.1630
.4511
.5831
.5854
.5878
.6035
.6382
.6312
.6211
.6237
.6216
.1576
.1991
.1979
.2079
.2137
.0467
.0458
.0432
.0508
.0486
10
9
9
9
9
= 0.980)
.0466
.0541
.0515
.0537
.0545
10
10
10
10
10
= 1.625)
.0986
.1259
.1228
.1243
.1263
--
a .- .6368, skewness
.0677
.0859
.0857
.0921
.0955
Runs
= 0.301)
a= .3087, skewness
.3134
.3063
.2963
.2989
.2969
= .5816,
.5647
.5830
.6082
.6020
.6079
8.101
4.677
5.773
13.628
8.243
a
fJ
.0762
.0987
.0973
.0955
.0976
.0479
.0542
.0522
.0545
.0561
10
10
10
10
10
= 2.475)
.0505
.0553
.0539
.0564
.0589
*For each distribution, 10 3PLN samples of l;ize 299 were generated.
10
10
10
10
10
e
125
Unfortunately, this analysis was conducted subsequent to the large
Monte Carlo
study
reported
in
Section 4.1.
·Had
it been conducted
sooner, the first quantile point used in the creation of grouped data
sets in Section 4.1 would have been set to 0.01 rather than 0.03.
126
A MONTE CARLO STUDY OF THE SMALL SAMPLE DISTRIBUTION
APPENDIX D:
e
OF EXP(aZ )
p
An important step in deriving the quantile regression estimation
(QRE)
equations
for
the
3-parameter
letgnormal
(3PLN)
distribution
involves the recognition that 3PLN quantiles are related by a monotonic
transformation to standard normal quantiles.
Thus, their expectations
may be computed in terms of the expectaticms of corresponding standard
normal quantiles.
While the exact expectations could not be expressed
in closed form, it was shown in Chapter 3 that a closed-form expression
did exist for the asymptotic expectations of the 3PLN quantiles.
were
computed
in terms
of
the moments
elf the
limiting multivariate
normal distribution of the standard normal quantiles.
this
Monte
Carlo
study
is
These
The purpose of
to assess how accurately the asymptotic
moments represent the exact moments of 3PloN quantiles for small sample
sizes.
Let 0 < PI < P2 < .•• < Pk < I, and let Xp l' Xp2 '
corresponding
sample quantiles
from a
... , Xpk be k
3P'LN distribution.
Using the
relationship between the 3PLN and standard normal quantiles
X .
p~
= y + exp(~)exp(aZ
(D. I)
.),
p~
it can easily be shown how the expectations of X . depend upon the first
p~
two moments of exp(aZ .).
p~
E(X .)
p~
Indeed,
=E[y +
=y
exp(~)exp(aZ
.)]
p~
+ exp(~)E[exp(aZ .)]
p~
(D.2)
e
127
and
Var(X .)
p1
=Var[y + exp(~)exp(azp1.)]
=
exp(2~)Var[exp(aZ
(D.3)
.)].
p1
Since the limiting distributions of the Z . are normal with means
p1
c.1
= ell
-1
(p .)
1
and
variances
k..
11
= [p.1 (l-p 1. ) ]Inep 2 (c.),
1
the
limiting
expectations and variances of the exp(aZ ;) can be derived by treating
p1
them as moment generating functions of normal random variables.
The
resulting asymptotic expectations are
lim E[exp(aZ .)]
n~
p1
=n~
lim exp(ac.
.
1
2
+ 1/2a k .. )
(D.4)
11
=exp(ac.)
(D.S)
1
since
lim k ..
n-+CO 11
= O.
The variances, of course, converge to zero.
lim Var[n ~exp(aZp1.)]
___
&.\.'-
= lim
n-+CO
tends to a finite limit.
However,
2k .. )[exp(a2k .. )-l], (D.6)
n exp(2ac.)exp(a
1
11
11
Rather than computing this limit, it was
evaluated numerically by computing the right-hand side of Eq. (D.6) for
larger and larger values of n until near convergence was attained.
The purpose of this Monte Carlo study is to compare the empirical
means and variances of exp(aZ .) with their asymptotic expectations when
p1
the sample size is small (between 19 and 99).
The divergence of the
empirical means and variances from their asymptotic expectations will,
apart from sampling variation, depend upon the value of a, the quantile
point p., and the sample size n.
1
Therefore, these three factors are
128
varied in this study.
The various values of a, p. and n for which the
1.
sampling behavior of exp (aZ .) was investi.gated are shown in Table D.1.
p1.
The four values of
a shown in Table D.1 are those that were used to
define the four 3PLN distributions studied by Cohen and Whitten [1980]
(and again in Chapter 4 of this paper).
The Monte Carlo study was conducted
:CIS
follows.
For each combin-
ation of nand p., 10,000 random samples lof size n were generated from
1.
the standard normal distribution.
Values of n were selected in such a
way that for each p. in Table D.1 there wlould be an order statistic in
1.
the sample whose cumulative distribution function had p .. as its expected
1.
value.~
Next, the p.th quantile, Z ., was computed for each sample and
1.
p1.
then the corresponding four values of exp(az ) were computed for each
pi
of the four values of a.
In this way, random samples of size 10,000 on
exp(aZ .) were constructed for each of 140 combinations of
p1.
a,
p. and n.
1.
For a given combination of p. and n, the samples corresponding to the
1.
four a-values were formed from the same sample of Z . so that comparp1.
isons of the sampling behavior of exp(aZ .) across the range of
p1.
a would
be more controlled.
To complete the Monte Carlo study, the first four sample moments
were
calculated for
each of the
140
samples
of quantiles and these
moments were compared to their asymptotic e:lCpectations.
asymptotic distribution of exp(aZ .) is given by Eq.
p1.
The mean of the
(D.S).
The expres-
sion for the asymptotic variance of n ~exp(aZ .) is given in Eq.
p1.
(D.6).
~A sufficient condition for such a value of n given a particular p. is
given in section 1 of Chapter 4.
1.
•
129
·e
Table D.1.
Values of
0,
p., and n Used in the
l.
Monte Carlo Study of Exp(oZ .)
pI.
e
0
p.I.
n
.0998
.05
19
.3087
.10
39
.4724
.20
59
.6368
.50
79
.80
99
.90
.95
130
w'hile all of the first four moments: are examined in this study,
only the first two moments must closely approximate their asymptotic
expectations
for the QRE procedure to be accurate.
Results for the
third and fourth moments are given only t,o assist the curious reader in
determining how nearly normal are the small sample quantiles.
The raw
results of the Monte Carlo study are shown in Tables D.2-D.5, in which
the means,
variances,
given, respectively.
skewness,
and kur'tosis of the 140 samples are
To aid in appreciating the effects of n, p. and cr
1.
In Figure D.I, for
on the approximation, two sets of plots are given.
the seven values of p., the percentage differences between the empirical
1.
and asymptotic values of E[exp(crZp.)] are plotted as' a function of the
1.
sample size for each value of cr.
for
Figure D. 2 presents analogous plots
Var[n~exp(crZp1..)].
Several
patterns
emerge
in each o:f
the
figures.
The notable
patterns are summarized by the following observations:
(a)
The
empirically
E[exp (crZ .)]
p1
and
determined
pe:rcentage
lim exp (crZ .) were,
p1.
~
"differences
between
holding other factors
constant, closer to zero when
(b)
1.
n was larger,
2.
P was nearer .20, and
3.
cr was smaller.
The
empirically
determined
1
pelt"centage
differences' between
1
Var[nllexp(crZ .)] and l!m Var[n llexp(crZ .)] tended to be closer
n
p1.
p1.
to zero, holding other factors constant, when
1.
n was larger,
2.
P was near .50, and
3.
cr was smaller.
'.
•
131
-e
Table D.2.
Sample Size
n
Distribution 1
19
39
59
79
99
lID
Distribution 2
19
39
59
79
99
lID
e
Distribution 3
19
39
59
79
99
lID
Distribution 4
19
39
59
79
99
lID
p.=.05
1
p.=.10
1
Means of Exp(oZ .)
p1
p.=.20
p.=.50
P 1.=.80
p.=.90
P 1.=.95
0.9159
0.9168
0.9174
0.9176
0.9178
0.9195
1.0001
1.0002
1.0000
1. 0003
1.0001
1.0000
1.0936
1. 0910
1.0902
1.0895
1.0895
1.0876
1.1492
1.1425
1.1414
1.1401
1.1393
1.1365
1.2018
1.1903
1.1864
1.1844
1.1833
1.1784
0.7648
0.7657
0.7668
0.7671
0.7674
0.7712
1.0032
1.0020
1.0011
1. 0016
1.0009
1.0000
1.3239
1.3115
1. 3078
1.3047
1.3046
1.2966
1.5458
1.5137
1.5081
1.5019
1.4986
1.4854
1. 7815
1.7212
1.7011
1.6913
1.6856
1. 6618
0.6661
0.6659
0.6669
0.6672
0.6675
0.6720
1.0081
1.0046
1.0027
1.0032
1.0020
1.0000
1.5431
1.5175
1.5099
1.5039
1.5035
1.4881
1.9598
1.8913
1.8787
1.86'63
1.8593
1.8321
2.4457
2.3068
2.2615
2.2398
2.2273
2.1754
0.5816
0.5797
0.5803
0.5803
0.5805
0.5852
1. 0153
1.0083
1.0051
1.0055
1.0036
1.0000
1. 8052
1.7594
1.7459
1.7358
1.7346
1.7089
2.4982
2.3702
2.3457
2.3236
2.3109
2.2619
3.3882
3.1063
3.0165
2.9744
2.9501
2.8510
1
1
1
(0 = .0998)
0.8340
0.8412
0.8439
0.8451
0.8456
0.8486
0.8716
0.8752
0.8769
0.8777
0.8778
0.8799
(0 = .3087)
0.5750
0.5879
0.5931
0.5953
0.5961
0.6018
0.6573
0.6639
0.6673
0.6688
0.6689
0.6732
(0 = .4724)
0.4329
0.4455
0.4509
0.4531
0.4539
0.4597
0.5294
0.5358
0.5394
0.5411
0.5411
0.5458
(0 = .6368)
0.3276
0.3382
0.3431
0.3450
0.3456
0.3508
0.4278
0.4329
0.4363
0.4378
0.4376
0.4421
..
132
Table 0.3.
Sample Size
n
p.=.05
1
Distribution 1
19
39
59
79
99
co
co
0.0333
0.0313
0.0316
0.0309
0.0314
0.0321
(0
co
co
p.=.20
P1.=.50
p.=.80
p.=.90
p.=.95
0.0171
0.0170
0.0170
0.0169
0.0174
0.0172
0.0156
0.0155
o 0158
0.0158
0.0156
0.0156
0.0253
0.0243
0.0242
0.0242
0.0241
0.0240
0.0401
0.0375
0.0366
0.0390
0.0382
0.0376
0.0733
0.0675
0.0638
0.0624
0.0613
0.0618
0.1136
0.1133
0.1137
0.1132
0.1165
0.1157
0.1510
o.lj~97
0.3605
0.3390
0.3363
0.3333
0.3329
0.3270
0.7097
0.6397
0.6180
0.6515
0.6358
0.6147
1.6203
1.3959
1.2785
1.2342
1.2079
1.1765
0.2017
0.2005
0.2018
0.2007
0.2065
0.2057
0.3589
0.3527
0.3585
O. 3~592
0.3531
0.3505
1.1659
1.0698
1.0565
1.0402
1.0382
1.0085
2.7326
2.3696
2.2638
2.3664
2.3032
2.1900
7.5076
6.0513
5.3829
5.1331
5.0008
4.7214
0.2800
0.2761
0.2784
0.2762
0.2841
0.2834
0.6661
O. 6j~77
0.6559
0.6571
O. 6j~47
0.6370
2.9584
8.2942
2.6346
6.8673
2.5861
6.4733
2.5275 . 6.7027
2.5200
6.5019
2.4168
6.0652
27.7200
20.6585
17.7509
16.6891
16.1706
14.7363
1
1
1
0.0231
0.0225
0.0218
0.0229
0.0224
0.0225
0.1245
0.1233
0.1206
0.1268
0.1243
0.1263
0.lj~95
0.1524
0.1526
0.1502
(0 = .4724)
0.1942
0.1918
0.1999
0.1964
0.2003
0.2108
Distribution 4
19
39
59
79
99
1
1
= .3087)
0.1481
0.1439
0.1483
0.1441
0.1482
0.1543
Distribution 3
19
39
59
79
99
p.=.10
n;~xp(oZp1.)
(0 = .0998)
Distribution 2
19
39
59
79
99
Variances of
(0'
0.1881
0.1878
0.1844
0.1940
0.1905
0.1944
e
= .6368)
0.2010
0.1997
0.2098
0.2062
0.2103
0.2230
0.2230
0.2227
0.2192
0.2306
0.2266
0.2317
•
133
Table D.4.
Skewness of Exp(aZ .)
p~
Sample Size
p.=.05
n
~
Dis tribution 1
19
39
59
79
99
(a
0.1203
0.0141
0.0488
-0.0059
-0.0226
Distribution 3
19
39
59
79
99
Distribution 4
19
39
59
79
99
p.=.20
~
P ~.=.50
p.=.80
~
p.=.90
1
p .=. 95
~
-0.1190
-0.0684
-0.0532
-0.0862
-0.0251
-0.0358
-0.0530
0.0255
0.0072
-0.0100
0.0719
0.0426
0.0320
0.0567
0.0514
0.2560
0.1508
0.1907
0.0965
0.1065
0.3020
0.5180
0.2634
0.4591
0.3396
0.1971
0.1232 .. 0.2918
0.1509
0.3286
0.1642
0.0906
0.1470
0.1049
0.0814
0.2541
0.1671
0.1337
0.1467
0.1286
0.4823
0.2991
0.3107
0.1997
0.1984
0.5657
0.4418
0.3397
0.2461
0.2631
0.8678
0.7207
0.5278
0.4531
0.4786
0.3204
0.2025
0.2425
0.1817
0.1532
0.3983
0.2649
0.2138
0.2176
0.1894
0.6673
0.4182
0.4064
0.2813
0.2707
0.7857
0.5862
0.4537
0.3437
0.3518
1.1685
0.9507
0.6841
0.5841
0.6002
0.4783
0.3148
0.3391
0.2591
0.2256
0.5457
0.3639
0.2948
0.2892
0.2509
0.8622
0.5413
0.5045
0.3644
0.3439·
1.0232
0.7366
0.5709
0.4432
0.4421
1.5015
1.2130
0.8507
0.7207
0.7267
(a = .3087)
Distribution 2
e
~
= .0998)
-0.1864
-0.1940
-0.1261
-0.1531
-0.1621
19
39
59
79
99
p. =.10
0.1339
0.0984
0.0843
0.0406
0.0820
(a = .4724)
0.3559
0.1741
0.1843
0.1077
0.0853
0.3289
0.2284
0.1920
0.1401
0.1661
(a = .6368)
0.5930
0.3336
0.3199
0.2210
0.1931
0.5250
0.3590
0.3006
0.2406
0.2509
•
134
Table D.5.
Sample Size
n
Distribution 1
19
39
59
79
99
Distribution 2
19
39
59
79
99
Distribution 3
19
39
59
79
99
Distribution 4
19
39
59
79
99
p.=.05
1
p.=.10
1
Kurtosis of Exp(oZ .)
p1
p.=.20
p.:=.50
p.=.80
1
p.=.90
2.9531
3.0170
3.0907
2.9482
3.0142
3.0249
2.9866
3.9734
3.0226
2.9548
3.2014
3.0670
3.0780
3.0470
3.0490
3.1568
3.3730
3.1416
3.5048
3.1039
3.2368
3.0164 ' 3.2269' '
3.0886
3.3554
2.9797
3.0105
3.1239
2.9695
3.0276
3.1196
3.0258
3.0052
3.0559
2.9835
3.5033
3.2084
3.1934
3.1081
3.0930
3.5806
3.3670
3.2438
3.0986
3.1660
4.1762
4.2032
3.5787
3.4568
3.5853_
3.0941
3.0546
3.1867
3.0091
3.0594
3.2781
3.0943
3.0559
3.1024
3.0208
3.8873
3.3819
3.3222
3.1846
3.1486
4.1355
3.6344
3.4074
3.2043
3.2600
5.1891
5.0863
3.9658
3.7163
3.8327
3.2935
3.1419
3.2829
3.0698
3.1104
3.5164
3.1983
3.1301
3.1675
3.0715
4.4264
3.6193
3.4885
3.2881
3.2235
4.9473
3.9949.
3.6242
3.3493
3.3856
6.6608
6.4287
4.4845
4.0576
4.1485
1
1
1
p.=.95
1
(0 = .0998)
2.9999
3.0234
3.0609
3.0383
3.1452
(0
= .3087)
2.9494
2.9440
3.0182
2.9814
3.0897
(0
3.0175
2.9425
3.0085
3.0890
3.0042
= .4724)
3.1246
2.9846
3.0614
2.9881
3.0937
(0
3.0435
2.9474
3.0016
3.0905
2.9946
3.1499
3.0043
3.0591
3.1319
3.0402
= .6368)
3.4898
3.1134
3.1705
3.0385
3.1385
3.4192
3.1248
3.1502
3.2153
3.1019
•
e
e
e:
•
..
p-.OS
Q)
c:::
Q)
4-l
I'H
u
~
Q)
~
0
po
oM
'd
~
'H
'H
....
00
I~
Q)
'H
....
oM
-20
444~cl~
p-.80
20
Q)
~
~
Q)
~
~
~o
4-l
~
~
wi
~ 0
'H
OM
'd
'M
ott
~
N
,
~4~~~c;
n
0
I
..20 I
•
•
00
p-.90
..~~~
-
zo
Q)
....
0
.~
Q)
~
Q)
'H
'H
0
4""
~
-20
-,\q
n
,,4
4'
n
n
p-.S5
~
..
LEGEND
~,~
--'"'"::-~~
oM
I
-24
44~~~
cf}
a = .0998
)C
a = .3087
...----......---a..=....--.
.4724
a -.........
a = .6368
.. _--/It.
• --
N
44
•
I
H
ott
4
t o-t
'H
....
'M
00
-JIG
~~~,,~~
n
zo
u
c:::
Q)
N
n
~
•
•
N
-J.ZQ '
Q)
u
~
0
p-.50
ao
Q)
u
c:::
Q)
Q)
p-.20
ao.,
Q)
u
~
p-.l0
m
2.0
~,., ~"q
4'
n
Figure D.l. Percentage differences between empirical and limiting
. values of E[exp(oZPi)] for varying
values of n t Pi' and o.
I ....
w
VI
"'
l
u
,..
~-
•
p-.05
.
-...--
---..,
t"-.~
~
p-.10
20.,
20
GJ
GJ
GJ
~
g
g
U
GJ
J.l
~o
....
't:I
tI-l
N
-e ,
U
~~
.-
?;~
444~4jJ
J.l
~o
....
"t1
tI-l
N
-2S ,
~
~-
GJ
u
~
GJ
J.l
GJ
tH,o
tH
....
'0
p-.80
~ ...
e.,
GJ
u
g
J.l
GJ
tH o
tH
n
·M
"t1
444~q~
-25 ,
100
~
GJ
u
~
g
J.l
GJo
tH
tH
'0
N
N
",,~~
n
~
....
~
-'IQQ
~
t-'1
r--"
......
p-.50
GJ
U
g
J.l
~o-l
....
'0
tI-l
•
-.
,-
N
44
4~q,fJ
-a
44tJ},,~~
n
n
p-.90
;
~
D
N
....
'0
-a
~
tI-l
n
orf
N
J.l
~o
~
p-.20
U
n
1.5.,
~
:---1
.p-.95
~tr
LEGEND
A
Mil
D
.
&IiJ ,,~
"q
)C
a = .0998
a = .3087
a = .4724
-------..
a = .6368
...-...---------------
• ----.._--~
n
Figure D.2. Percentage differences between empirical and limiting values of Var [n~exp(aZ i)] for
...varYing valuea of n, Pi' and o.
...
P
I
.....
w
4It
0\
•
~