•
ON THE USE OF PRELD-ilINARY TESTS IN REGRESSION
Vijay Govindji Ashar
Institute of Statistics
Mimeograph Series No. 698
July, 1970
A.~ALYSIS
ABSTRACT
ASHAR, VIJAY GOVINDJI.
Analysis.
On the Use of Preliminary Tests in Regression
(Under the direction of THOMAS DUDLEY WALLACE).
This is a problem dealing with the implications of using a
preliminary test on subsequent estimation of a regression parameter(s).
By a preliminary test is meant here a test on a nuisance parameter which
precedes estimation of the parameter of interest.
Two different proce-
dures for the use of a test are compared for efficiency in the final
estimation.
One procedure involves dependence of the ultimate estimator
on the outcome of the preliminary test, the resulting estimator being
called the TE estimator.
The second procedure is characterized by one-
to-one functional dependence on the test--statistic itself resulting in
the continuous weight function (CWF) estimator.
All the results are
illustrated with the linear regression model containing two nonstochastic
regressors.
Given the objective of obtaining a "good" estimator of 6 ,
1
two somewhat interrelated situations, viz., that arising from specification error, and one from multicollinearity, are envisaged.
Both these
situations, however, can be handled with the same testing procedure, with
only a change in the significance level (a), one corresponding to the
usual t-test, and the other to the MSE test.
The central objective of this investigation is to examine, first
analytically, the influence of the various parameters involved on the
bias and MSE of both the TE estimator and the CWF
estima~or;
subsequently,
via a comparative analysis of some numerical results, to study the
relative degree to which each estimation procedure can control the MSE
over the range of the nuisance parameter.
From a combination of analytical results and numerical observations,
the following conclusions of major significance emerged:
(i) The bias in both the TE estimator and the CWF estimator displays
similar pattern of behavior over the range of the (unknown) nuisance
parameter (denoted by A):
It increases first, reaches a maximum, and
then steadily decreases to approach zero, as A approaches infinity,
However, in any comparable situation of the two estimation procedures
(!.~.,
after normalization in some sense), both the rate of increase and
the rate of asymptotic convergence to zero are different for the TE
estimator and the CWF estimator.
(ii) The MSE of the TE estimator lies between the MSEs for the two
simple estimators, viz., b
l
and 8 , over the two extremal ranges of the
1
nuisance parameter A, say, [O,A
O
]
and [Al,oo].
However, for every value
of a not equal to zero or one, there exists an intermediate range of A
over which the MSE of the TE estimator is larger than both MSE(b ) and
l
MSE(8 ).
l
Thus, over this range of A, the TE procedure is less desirable
in the MSE sense than even an arbitrary choice of b
l
or 8 as the
1
estimator of 8 .
1
(iii)
On
the other hand, for any typical member of an important
subclass of the general CWF estimator of 8 , there exists an intermediate
1
range of A over which its MSE is smaller than both MSE(b ) and MSE(8 ).
l
l
Thus, if a large prior weight can be attached to intermediate range of
A (as against the two extremal ranges), then the choice will obviously
be in favor of the CWF estimation procedure.
(iv) Further, numerical results showed that on the whole, the CWF
procedure (for the exponential family of weight functions) exercises a
distinctly greater degree of control on the MSE, compared with the TE
procedure, over the entire range of A.
ON THE USE OF PRELIMINARY TESTS
IN REGRESSION ANALYSIS
•
by
VIJAY GOVINDJI ASHAR
A thesis submitted to the Graduate Faculty of
North Carolina State University at Raleigh
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
DEPARTMENT OF EXPERIMENTAL STATISTICS
RALEIGH
1 9 7 0
APPROVED BY:
Chairman of Advisory Committee
ii
fI
BIOGRAPHY
The author was born on December 24, 1939, in Bombay, India.
He
graduated from Kabibai High School in 1955, received a B.A. (Hons.) in
j
Statistics and Mathematics from Bombay University in 1959, and launched
his educational career in the United States as a graduate student of
mathematics at the University of Iowa in Fall, 1959.
He transferred to
the then North Carolina State College at Raleigh in the Department of
Experimental Statistics in Spring, 1960, and obtained the M.S. degree
in Experimental Statistics in August, 1961.
Subsequently, he worked as
an analytical statistican with Cornell Automotive Crash Injury Research
in New York City, and returned to India in September, 1962.
While in
India he worked as a statistician in the Economic Intelligence Unit of
a major industrial corporation in Delhi until he returned to North
Carolina State University at Raleigh in Fall, 1964, for pursuing a Ph.D.
program.
He was concurrently appointed a Research Assistant from 1964
to 1968 jointly with the Department of Statistics and Department of
Economics,and went back to India in November, 1968,after completing
his Ph.D. requirements.
He also spent the summer of 1966 with the
Mathematics Research Division of the U. S. Steel Applied Research
Laboratory in Monroeville, Pennsylvania.
While in India, the author married Charu Bhimani of Bombay in
December, 1968, and returned with his bride to the United States in
January, 1969, to take up a position with International Business
Machines in East Fishkill, New York.
He has been with the same group
since then, and was recently promoted to Staff Statistician.
iii
ACKNOWLEDGMENTS
The author wishes to express his deep appreciation to T. D. Wallace,
Chairman of his Advisory Committee, who has been a constant and incorJ
rigible source of inspiration and guidance throughout his Master's and
Ph.D. programs, and to B. B. Bhattacharyya, who has contributed many a
valuable suggestion and otherwise offered his impeccable guidance to
improve the final quality of this dissertation.
The comments and sugges-
tions of the other members of his committee, H. R. van der Vaart and R. A.
Schrimper, are also sincerely appreciated.
Special gratitude goes to Mrs. Jo Ann Beauchaine for a devoted and
excellent job of typing the earlier versions as well as the final draft
of this thesis, replete with heavy and involved formulae.
The author is also grateful to Mrs. Teddy Kovac for drawing attractive
charts, and to James Barr for his help in writing and debugging computer
programs used in numerical calculations.
The author feels particularly indebted to R. L. Anderson, now at
the University of Kentucky, for having shown a continual interest in his
educational and professional aspirations, and to W. D. Toussaint for the
encouragement the author received during his Ph.D. program.
A word of
gratitude is also due to the Department of Economics and the Department
of Experimental Statistics for the provision of financial support which
enabled the author to carry out his study program.
Finally, the author acknowledges with fondness the well-meaning
proddings of his wife, which definitely hastened the completion of this
project.
iv
TABLE OF CONTENTS
Page
LIST OF TABLES .
v
LIST OF FIGURES
vi
INTRODUCTION • •
1
Statement of the Problem •
Review of Literature . • .
1
4
THE TWO-STAGE ESTIMATION PROCEDURE •
6
Mathematical Background. • • • •
The Two-Stage (TE) Estimator • •
Distribution and Moments of the TE Estimator •
Some Properties of the Q Function.
General Observations about Relative Bias and Relative MSE. •
Comments on MSE. • • • • • • • • • •
CONTINUOUS WEIGHT FUNCTION ESTIMATOR •
The Family of Exponential Weight Functions • • • • • . • • •
Some General Comments on Bias and MSE of Z •
e
Comments on MSE. • • • • • • • • • •
Normalization of Weight Functions. • • • • • •
• • • • •
Derivation of Expectations Occurring in the Moments of Z
e
NUMERICAL RESULTS AND COMPARISONS. •
Summary. .
. . .
. .
.
23
24
25
27
29
30
32
35
38
42
48
48
Choice of Parameter Values •
Comparison of Relative Bias. •
Comparison of Relative MSE •
Conclusions. • • • • •
• ••••
Suggestions for Further Research
15
17
18
48
Prologue . • . . . . . . . . .
SUMMARY AND CONCLUSIONS.
8
11
23
Introductory Discussion. • • .
Definition of the Class of CWF Estimators and Its Moments ••
Selection of a Weight Function Based on an Optimality Criterion. •
An Optimal Parametric Weight Function. • • •
• • • .
Some "Reasonable" Estimators of .Q,~ • • • • •
•••••••••
•
6
51
59
····
····
····
75
75
76
77
LIST OF REFERENCES
79
APPENDIX • • • • • •
81
v
LIST OF TABLES
Page
1.
Z.
3.
4.
Values of E(o ) for selected values of A, a and v compared
With!~
u
• • • • • • • •
• • • • • •
43
Values of E(~e) for selected values of A, ~ and v compared
With!~
u
• • • • • • • • •
44
Values of the noncentrality parameter A for selected
combinations of 6 ' p and v • • • • • • •
Z
Values of
p~z E(Sl-Ze) for selected values of SZ'
p
50
and
v = 2 • • • • • • • • • • • • • • • • • • • •
5.
Values of
-iE(Sl-Z )
p""Z
e
~, and
• • • •
for selected values of SZ' p and ~, and
\) = 8 • . • • • • • • • • • . • • • • • • • • • • • • • • • •
6.
Values of
p~z
E(Sl-Ze) for selected values of SZ'
\) = 18. . . . .
7.
8.
. . . . .
Z
10.
11.
lZ.
13.
~,
54
and
. . . . . . . .
*
Values of the ratio MSE(Z!)/MSE(b
l ) for selected values of SZ'
55
58
..... . . . . . . .
60
Values of the ratio MSE(Ze)/MSE(b ) for selected values of A
l
and .;, \) = 2. . .
. . . ..
. . . . . .
61
Values of the ratio MSE(Ze)/MSE(b ) for selected values of A
l
and E.:, 'V = 8. . .
. . . ..
..... . . . . . . .
63
Values of the ratio MSE(Ze)/MSE(b ) for selected values of A
l
and ~, \) = 18 . .
. . . .
. . . . . . . . . .
65
. . . ..
*
1 and Z! for selected values of A over
Relative MSEs of Ze' 6
the range of values of p:
•
and
1
p and \)
9.
p
Values of __
1 _ E(S -T) for selected values of SZ' p, v and a
p6
53
The collinearity effect • • • • • •
Values of the ratio MSE(T)/MSE(b ) for selected values of A,
l
'V and ex.
• • • • • • • • • • • • • • • • • • •
72
73
vi
LIST OF FIGURES
Page
1.
Typical MSE(Q) curves of b , Sl and T for fixed p and v •
l
a
2.
Relative bias of Z for different
e
3.
Relative MSE (Qz /Q ) curves for different .; values and
b
e
1
p = .40, v = 8
~,
and v
=8
•
···············
4.
Relative MSE (QZ /Q ) curves for different c; values and
b
e
1
p = .60, v = 8
············· ··
5.
Relative MSE (Qz /Q ) curves for different .; values and
b
e
1
p = .90, v = 8
...···············
,
.
20
56
····
67
····
68
····
69
CHAPTER I
INTRODUCTION
Statement of the Problem
This thesis aims to analyze the implications of using a preliminary
test on final estimation of a regression parameter (or a vector of
parameters).
By a preliminary test is meant here a test on a (nuisance)
parameter(s), which precedes estimation of the parameter(s) of interest.
Two rival approaches to the use of a test are proposed and compared for
efficiency in the final estimation.
One procedure, described inter-
changeably as the two-stage estimation procedure or the test-estimation
(TE) procedure, involves dependence of the ultimate estimator on the outcome of the preliminary test; the second procedure, called the continuous
weight function (CWF) estimation, is characterized by one-to-one functional dependence on the test statistic itself.
Suppose we have data for a sample from the following linear regression
model with two nonstochastic regressors, satisfying the usual normality
assumptions:
Suppose further that our primary objective in analyzing the data is
to obtain a "good" estimator of
~.
..
al •
(This implies that we focus on the
problem of structural estimation, as against one of prediction.)
Obviously, the question of which is a "good" estimator will depend upon
the criterion of goodness chosen.
In most practical applications of this model, especially in social
and behavioral sciences, two different situations, not completely
2
unrelated to each other, arise:
variable x
2
(i) We are not quite sure whether the
is pertinent in explaining the behavior of y,
S2 is significantly different from zero.
i.~.,
whether
The standard procedure to
handle this uncertainty is to make a test on the significance of S2' and
subsequen~ly, to estimate Sl by either the ordinary least squares (OL5)1
estimator b
1
or the restricted least squares (RL5) estimator Sl' depending
on the outcome of the test.
(ii) Often, the undesigned variables xl and
x ' although individually quite significant in explaining the behavior of
2
y, are themselves highly intercorre1ated.
case of serious co11inearity
(i.~.,
It is well known that in this
very high value of p), the OL5 esti-
mator of Sl' although still unbiased, will become less and less reliable,
its variance fast approaching infinity as
a simple way out, we may want to drop x
Sl) even when S2 is not, in fact, zero.
2
p2
approaches one.
Hence, as
(for the purpose of estimating
If we adopt the criterion of
minimizing the M5E (which equals variance + (Bias)2) to choose a "good"
estimator, it turns out that we must use the same test-statistic as in
(i) for deciding between b
1
and Sl' although now with a different range
of the null hypothesis.
Note that in both the above situations, we are operating in the
framework of an incompletely specified model in that the choice of which
model should be used for the ultimate purpose of estimation is based on
the outcome of a preliminary test on a "nuisance" parameter.
1
We will
The OL5 estimator b 1 corresponds toAthe "full" model; i.~., including
both xl and x2' while the RL5 estimator Sl refers to the model incorporating the zero restriction S2=0. The reader is urged to get familiar with
£his abbreviation, and keep in mind the proper distinction between b and
1
Sl throughout the pages of this thesis.
3
call the estimator resulting from this two-stage procedure the TE
estimator, to be denoted by T.
In Chapter II, we shall derive the den-
sity function and the moment-generating function of T, and investigate
its bias and MSE in considerable detail.
The TE estimator can be described as a weighted sum
combination) of the simple estimators b
l
(.!..~.,
convex
and Sl' the weights being zero
or one depending on whether acceptance or rejection of the respective
hypothesis is prescribed by the preliminary test.
This step-function
weighting procedure implicit in the TE estimator can be generalized to
the procedure involving a continuous weight-function (again, as a function of the preliminary test-statistic).
In this latter case, we do not
carry out a test as such, but merely use the test-statistic as a "proxy"
for the unknown parameter which determines its distribution.
As
we
shall see in Chapter III, ideally the weight function in the convex
combination of b
l
and Sl should be expressed as a continuous function of
the same (unknown) parameter.
Hence, by a judicious choice of an esti-
mator for this parametric function, we can expect to improve upon the
two-stage estimation implied in the TE estimator.
In Chapter III, we
shall study the various possibilities in this direction, and in
Chapter IV, we shall specifically investigate, via a comparative study
of some numerical results, whether the continuous weighting procedure
on the whole exerts a greater degree of control on the bias and MSE of
the ultimate estimator of Sl.
4
Review of Literature
The problem of estimation after a preliminary test of significance
has been studied for different situations by a number of investigators.
Some of them have examined both bias and MSE of the resulting TE estimators, although not exactly in the same framework, and generally not with
as detailed perspective as the one presented in Chapter II.
Bancroft
(1944) first formally pointed out the implication of the test-estimation
procedure for the same model being considered in this thesis.
He also
derived and examined in some detail the bias introduced in the estimation
(of Sl) preceded by a t-test (for the significance of S2).
Mosteller
(1948), Kitagawa (1950) and Bennett (1952) all discussed the application
of the TE procedure in the classical two-sample pooling problem for
estimating the mean of a normal population.
Bancroft (1944) considered
the analogous problem for the estimation of variance in a normal
population.
Kitagawa (1950) also considered the problem of "sometimes
pooling" in interpenetrating sampling, where actually one has a sample
from a bivariate normal population with means
wants to estimate
~.
x
~
x
and
~
y
, say, and one
Kitagawa (1963) discussed the use of TE procedures
in response surface analysis.
Larson and Bancroft (1963) and Kitagawa (1963) studied both bias and
mean square error (MSE) for the predicted value of ybased on a pre1iminary t-test for the significance of a subset of regression parameters
:
in the general multiple regression model.
Recently, Sawa (1968) treated
the same problem with somewhat more elegant mathematical formulation.
For the two-regressors linear model considered by Bancroft (1944),
Wallace (1964) proposed and derived the MSE criterion, discussed in the
5
previous section, for making a choice between b
1
and Sl in the context of
high intercorre1ation between the regressor variables.
Toro and Wallace
(1968) generalized this criterion to the general linear model and to the
case of nonzero linear restrictions.
They also derived a UMP testing
procedure for applying the MSE criterion in the generalized model.
Toro
(1968) obtained the bias and MSE expressions for the TE estimator of Sl
based on the MSE test, and tabulated some numerical results, part of
which have been reproduced in this thesis for illustrative purposes.
Gun (1965) made an exhaustive study of the effects of the TE procedure on
the bias and MSE of the predicted value of y in a factorial model when the
MSE criterion is used to determine whether or not to include the interaction term.
The generalizations on bias and MSE of the TE estimator
presented in the next chapter generally run parallel to, and in some
instances, are extensions of related comments in the studies of Bancroft
(1944), Toro (1968) and Gun (1965).
Huntsberger (1955) proposed a generalization of the TE procedure in
the form of a convex combination involving a continuous weight function
of the test-statistic.
He also made a comparative study of the two
procedures in the two-sample pooling problem with the test-statistic
following a normal distribution.
His results did, indeed, show that the
continuous weighting procedure on the whole effects a greater degree of
control on the MSE than does the TE procedure.
Kitagawa (1963) and Bancroft (1964) have brought together many of
the findings surveyed above in the respective review articles.
Of these,
the former is the more wide-ranging one, where the author has discussed
some other aspects of the TE procedure as well.
6
CHAPTER II
THE TWO-STAGE ESTIMATION PROCEDURE
Mathematical Background
Consider the two-regressors linear model:
•
(1)
Apart from the usual least squares assumptions, we assume without
loss of generality, that SX
i
= Sy = 0
(i
= 1,
= p,
=
2), so that
with -1 .:.. p < 1
where S denotes summation over the sample observations, [a
ij
] is the
sample moment matrix of the x-variables and p is a measure of the
intercorre1ation between the regressors.
We also assume a normal error process, so that all the regression
estimators in the following discussion, being linear functions of normal
variables, are normally distributed.
The least squares estimators for
the regression parameters in model (1) are:
i
= 1,
(2)
2
with
(3)
where
i = 1, 2;
and
02
is the common variance in the distributions of
E.
7
If, however, we assume S2 = 0, we will use the model
(4)
The estimate of Sl in this case is given by:
(5)
with
a
E(Sl) = S +
1
~
S
Var (Sl) =
all 2'
(6)
It can be easily shown that
(i)
(7)
(ii)
(8)
(iii)
COV (Sl' b ) = 0,
2
(of errors), Sl and b
2
!~~.,
under the assumption of normality
are stochastically independent.
In the absence of any
~
priori specification about S2' the usual
procedure for testing the significance of S2 is to use the following F
statistic from the standard ANOVA table:
a
2
2
b S(x - -12
2
x
)
RSS(b )
2
b
2
all 1
2
2u =
=
=2
2
EMS
S(y-x b - x b ) / (n-3)
c s
2 2
1 1
22
where c 22 = a
u
~
1
2
(1-p2) , and s is the EMS.
It is known that
22
° <=> A = 0, and
for any member of the alternative H* : S2 +° <=> A > °
a
F(1,n-3=v) under the null hypothesis H* : S2 =
o
u ~ F , (l,v;A)
where A, the noncentrality parameter, is given by:
(9)
8
(10)
The Two-Stage (TE) Estimator
If, as described in the previous chapter, specification error in
•
model (1) is our primary source of concern, and if no independent
~
priori
information on possible values of Sz (or, equivalently, on A) is available,
then the choice between models (1) and (4) must be based on the result of
the preliminary test of H*' and the TE estimator of Sl takes the form:
o
b
T* = {
1
if H* : A
0
0 is rejected
Sl if H0* : A = 0 is accepted
(11)
On the other hand, if intercorre1ation between the regressors poses
a serious problem, then it would be more appropriate to use the MSE
criterion for a choice between b
1
and Sl'
Z
in which case the preliminary
test should be based on the hypothesis:
1
Z
H : A <
o
ZChoosing the MSE criterion to determine the estimator of Sl implies
that we no longer impose the restriction of unbiasedness on the class of
estimators from which we pick out the optimal one. Thus, under this
criterion, the choice between b1 and 61 remains meaningful even when Sl
becomes biased (!.~., when Sz
0). Since, in this latter situation,
it would take increasingly higher values of S~ (contributing to the bias
A
+
A
A
of Sl), as p2 approaches unity, to make Sl less preferable than b1' the
MSE criterion reflects the degree to which co11inearity among the XIS
distorts the conclusions based on classical least squares analysis.
Gun (1965) has classified the test based on the MSE criterion as the test
for "material significance" (of SZ) to be distinguished from "ordinary
significance" implied in the usual ANOVA test.
9
!.~.,
the TE estimator will take the form
b
ifH : !..
l
0
1
<
2
T = {
13
•
1
1
i f H : !.. <
-
0
2
is rejected
(12)
is accepted
*
Note that the same test-statistic u is used for both Hand
H.
o
0
But in
the latter case, u will take noncentral F distribution for any general
member of the null hypothesis as well.
In fact, since both F and F
,
distributions satisfy the monotone likelihood ratio property, the liMP
*
tests for Hand
H will be identical except for a change in the level of
o
0
significance.
Hence all the results will be obtained in the context of
the preliminary test on H , with the understanding that they will be
o
applicable, mutatis mutandis, if the test on H*
o
(!.~.,
the usual ANOVA
test) is used instead.
Denoting the level a rejection region for H
o
(in the sample space or,
C
equivalently, the u space) and its complement by Rand R respectively,
a
a
the TE estimator can be written as a step function of the preliminary
test-statistic u as follows:
T = 0ubl
+ (1 - °u H\
=
(31 + 0u (b l -(3l)
(13)
where
°u = 1 if u
£
R
a
a
£
R
a
°u =
Here, obviously, R
a
point in the F' (l,v;!..
expression:
if u
=
[u >
-
= ~)
U
a
],
where u , the upper 100 a percent
a
distribution, is explicitly given from the
10
00
a
=f
u
(l,v;A
d F'
= 1:.)
Z
(14)
a
It will be more expedient to use the Beta transformation of u,
defined by:
(l/v)u
w = l+(l/v)u
=
u
v+u '
with 0 < w < 1
(15)
It is well known that w takes the noncentral Beta distribution,
written succinctly as w
~
Be
,
1 v
(2'
2;
A), where A is the same noncentrality
parameter (10) as in the distribution of u.
Further, the density function
of w can be expressed in terms of the following infinite series (see, for
example, Graybill (1961), p. 79):
00
g(w) =
~
[P{pr(x=m; A)}]
m=o
where the terms in the first
[Be(w;~,
r)]
(16)
] denote successive probabilities from a
Poisson distribution with parameter A, and those in the second
characterize respective Beta densities with parameters (m+f'
f ).
u
Since [u ,oo]<=>[w ,1], w
a
a
a
1
-
1 - a
where I
.
w
= vttia
satisfies the relation
. a
00
= eZ ~
1
m=o Zmml
Iw
a
1 ~)
( m + 2'
Z
is the corresponding incomplete Beta function ratio, with
a
(17)
11
Distribution and Moments of the TE Estimator
Using the result (13), the density of T can be written as follows:
g(t)
where gl and 8
2
= gl(t/u>ua )
c
• Pr[R ] + g2(t/u<u ) • Pr[R ]
a
- a
a
are the conditional density functions of b
l
(18)
and Sl
respectively.
Noting that u is a fn. of b
2
and s
2
only, and that Sl' b
2
and s
2
are
mutually independently distributed, the first two normally, as
a 12
Z
N(SI + all SZ' a 2 /a ll ) and N(S2' a 2 c Z2 )' and the last one as Xv a 2 /v ,
it can be shown that g(t) is given by
a
- N(SI +
X
where f
l
and f
2
a~~
2
SZ' a /a ll )]
f (b ) • fZ(sZ) dbZds
l Z
denote the densities of b
2
and s
2
(19)
Z
respectively.
Hence, we immediately have the following results for the expected
value and MSE of the TE estimator (they also follow directly from the
second result in (13»:
E(T)
(20)
where
(Zl)
a
E(T-Sl)Z
= MSE(T) = MSE(Sl) + a~~
p2[J 2 - ZS2Jl]
(ZZ)
12
where
(23)
The values of J
l
and J , and hence the first two moments of T about
2
Sl' can be directly evaluated by infinite series expansion of the integrals involved.
3
However, it will be more instructive to obtain
expression for the moment-generating function of T, and deduce the
required moments therefrom.
Using the identity (7), and the fact that Sl is independent of both b
2
and u, we obtain
m(S)
= E[exp{SlS}] (E[exp{-
a
12
(---)b S}lu>u ]·Pr[R ] + l-Pr[R ]}) (24)
all 2
a
a
a
It is easily seen from previous results that
Pr[R ] = Pr[u>u ] = Pr[F' (1,\);1..) > u ]
a
a
a
= Pr[Be
..
, 1 \)
(Z'Z;A) > w ]
a
=1 -
Q (A)
1
2
where
3
See, for example, the derivation of bias in Bancroft (1944) and of
MSE in Toro (1968).
13
,
00
Q (A) =
j
m=o
a
exp{- 4
(25)
12 f3 8
all
=
\)
Pr[Be (j '-2; A) -<w
]
(1,
2
a l2
exp{- f3 8
all 2
where
(26)
so that
Substituting these results in (24), we get
m(8)
(28)
.'
.
,
4Note that the argument of Q becomes A* because the normal random
l
2
variable in the double integral now has location parameter
f3 2 ·
~
instead of
14
The cumulant-generating function, log m(6), is given by
k(6)
=
[(a
a12
62
*
+ --- a )6 + cr 2c (1-p2)--2] + log·[Ql(A) + {l-Ql(A )}
all 2
11
1
2
2
(29)
In generating the cumulants from k(6), the following results, which
can be easily verified from (25) and (27), will be utilized:
,
= Qj(A) = Qj+l (A)
"
= Qj+2(A)
Qj(A)
Q'" j(A)
= Qj+3(A)
- Qj(A)
- 2Qj +1 (A) + Qj(A)
(30)
- 2Qj +2(A) + 3Qj +1 (A) - Qj(A)
Hence the mean of the TE estimator is given by
E(T)
=k
,
a
(0) = a
1
+ ~ a
all
2
(31)
Q3(A)
2
Comparing this with (15), we can immediately write down
= a2 [1-Q3(A)] = aZPr[w *>wa ]
.
'
where
w*~
(32)
2
Be
'3
v
(2' 2;
Also, Var(T)
A) •
2
= k " (0) = cr2c1l[1+p2{2A-1)Q3+2A(Q3-QS)-2AQ3}]
- 222
where the argument of Q and Q is A.
3
S
2
2
Thus,
-
2
(33)
15
a
12
Bias (T) = --- S Q3(A)
all
(34)
2
2
2
MSE (T) = Var(T) + Bias (T)
= cr 2c ll [1+p2{(2A-l)Q3 + 2A(Q3-QS)}]
-
-
(35)
,-
222
A
= MSE(Sl) - cr2cllP2[(2A-l) (1-Q )-2A(Q -Q )]
3 S
-3
222
A
a
22
= MSE(Sl) + --- p2(cr 2c
all
22
)[2A(Q -Q )-(2A-l) (l-Q )]
1.
~
1.
222
(36)
Comparing (36) with (22), it can be seen that
J 2-2S 2J l = cr 2 c 22 (1-Q3) + S~[(Q3-QS) - (1-Q3)]'
222
2
or
J
2
b
2 = E(Ou ;) = cr c 22 (1-Q3) +
S~(l-QS)
2
where w**'V Be 's
(2'
\I
'2;
(37)
2
A).
Some Properties of the Q FunctionS
6
.'
(i) Since Q
j
co
=
~
m=o
Pr(X=m) Pr(Ym+j
< W )
a.
SAlmost all these properties have been described in Gun (196S).
they have been included for the sake of completeness.
(38)
Here
6Henceforth, throughout this chapter, it will be understood that the
argument of Q is A unless otherwise mentioned.
j
16
where X has a Poisson distribution with parameter A, and Y + has a
mj
\I
central Beta distribution with parameters (m+j; 2)' it is obvious that
o -<
Q. < 1, the lower and upper bounds being attained only when w = 0
a
J -
and 1 respectively.
(w
= 1 <=> a = 0 => adopt restriction; w = 0 <=>
a
a
a = 1 => FLS model.)
(ii) Other things remaining constant, Q increases monotonically in
j
[0,1] as w increases from 0 to 1, or equivalently, as a decreases from
a
1 to 0,
\I
and A being fixed.
(iii) For 0
w
1, I
<
a
w
(m+j; ~) - I
a
w
(m+j+1; ~) =
a
\I
\I
_
<
f (m+j+:-2)
_
_ _.::..-_ _
\I
W m+j (1 -w )2 >0.
a
a
Hence
f(m+j+1)r(2 -1)
00
(l-w )2
a
Pr (X=m)gm+j+1 ,
\I
m=o
where gm+j+1
(2 -
2
~_ 1 (wa ) is the density of Be(m+j+1, ~
, 2
wa' and X is the Poisson r.v. with parameter A.
cally decreasing sequence in j.
(39)
\I
1)
1) at the point
Thus, Q is a monotonij
We also observe from (39) that Qj-Qj+1
is not a monotonic function of a (although Q. is).
J
(iv) From (30),
o
.:
i·~·
, Qj is a monotonic decreasing function of
Actually, by a lemma,
7
we have the result that
7
For a proof of this lemma, see Gun (1965).
A for 0 < w
a
< 1.
17
lim AaQ
1.-+00
(v) For A = 0, Q
j
=a
j
=
for a
>
a
=>
lim Q
j
=a
(40)
1.-+00
I
wCJ.
(j
,f) .
General Observations about Relative Bias and Relative MSE
Since the OLS estimator b
is unbiased (this would be true even when
l
the true model is (4) instead of (1»
it would be interesting to compare
+a in
the bias of T with that of Sl' when S2
p
+0,
the true model, and when
since the latter two cases result in zero bias for both T and Sl.
From (36), we see immediately that
Bias (T)
R. B. =
Bias (Sl)
Hence,
A
(i)
l.~.,
a
<
IBias(T)I
<
IBias(Sl)I
the bias of T always lies between that of b
l
and Sl' the relative
magnitude of the bias being equal to a certain probability in a noncentral
Beta distribution.
(ii) It follows immediately from property (iv) of the Q function
that the relative bias approaches
a
as A increases from
a
to 00, a fact
which is intuitively self-evident, since larger values of A correspond to
.. -
greater power for the UMP test, and consequently a larger weight to the
unbiased estimator b •
l
The upper limit on the bias for w
CJ.
from property (v) above to be
R. B. < I
w
CJ.
<
1 is given
18
It is seen from (10) that this upper limit is attained when p2
=1
(!.~.,
when A = 0).
(iii) For a fixed A and v, the relative bias decreases monotonically
in [0,1] as a increases from 0 to 1.
This follows from property (ii) of
the Q function.
(iv) The sign of the bias of T depends upon the signs of
p
and SZ; if
both have the same sign, the bias is positive; if both are of opposite
signs, the bias is negative, and the bias is zero is either of them is zero.
The bias of T always takes the same sign as the bias of Sl.
Comments on MSE
Throughout the following, we will use the notations nand MSE
interchangeably.
We first observe that
As seen before, n
b
1
= a 2 c ll •
Hence, we can rewrite (35) and (36) as
follows:
(41)
(42)
."
where k(A)
= ZA(Q3
Z
:
- Q5) - (1-2A)Q3
Z
k * (A)
= k(A) +
(1-2A).
Thus, we
Z
immediately get
(i)
= ca
k(A),
(43)
19
= co
(ii)
so that 0
<
*
k (A) > 0
c
o
<
nb .
l
With the help of these results, and using appropriate properties of
the Q functions, the following generalizations can be obtained on the MSE
of T:
(i)
For 0
nT
<
nT
<
<
nb
a
<
for
1,
1
8
4" .::..
1
1
A < - - ° 0'
-2
nT
nb
>
for A >
1
1
2
°0
(44)
and
n6
A
for
1
4" .::.
1
A .::..
1
'2 +
°1 ,
nT
<
n6
A
for A >
1
1
'2 +
°1 (44)
where 00 and 01 are strictly positive quantities which depend on a.
In
words, these results imply that the MSE of T can be smaller than one or
the other of the MSEs of b
l
and 6 , but never smaller than both of them,
1
for any particular value of A.
A
E(t - 00' t + 01)'
Further, even more ominously, for
!.~., for a certain interval of A in the neighbor-
hood of ~ (its length depending on the particular choice of a), MSE of T
A
..
is larger than both MSE (b ) and MSE (6 ).
l
1
Figure 1 illustrates these
points for typical MSE curves for given values of p, v and a •
8For a = 0 and 1 (1. e. ,w= 1 and 0), the TE estimator collapses to
--
a
6 and blrespectively. This fact is also verified through equalities
1
obtained for the respective results in (43).
•
e
-,
e
MSE
1
1
(2 °0' '2+°1)
I
I
I
I
I
I
I
5
:
(4'
.
5
·········
'4 + °2)
rl
b
1
I
I
I
I
I
I
I
I
I
o
I
I
. II
I
1.0
.5
Figure 1.
I
1.5
I
I
2.0
2.5
Typical MSE(rl) curves of b , Sl and T
1
a
,
I).
4.0
3.0
for~fixed p
and v
I'-)
o
21
(ii)
lim k(A) = O.
A-+oD
Hence
!..~.
,
for every a £(0,1) and for a fixed v and p2, the MSE of the TE estimator
approaches that of b
1
as A (and consequently S~) becomes very large.
This is again intuitively obvious, for the same reason provided in the
analogous property (ii) for the bias of T.
However, the rate of this
asymptotic convergence varies, not only for each a, but also for different
(iii) k (A) = 2(Q3-QS) - 2A(Q3- 2Q S+Q7) + 2Q 3 + (1-2A)(Q3-QS)
-2 2
222
2
2 2
>
2A(QS-Q7) - (4A-S) (Q3- QS)
2 2
2 2
S
>OVA<-
-4
Thus, for fixed values of v and p2, and every a £ (0,1), Q is a monotonic
T
increasing function of A
(!..~., ofS~) in the range [0, ~ + 02]' where 02
is a positive quantity, determined by the particular choice of a.
The
S
maximum of Q occurs at some point to the right of A = 4 ' beyond which
T
drops back steadily to converge asymptotically to Q •
b
1
3 v
2
(iv) For A = 0 (i.~. , for S2 = 0), R = 1-p r
which is
w (2' 2)'
a
monotonic increasing in a. Thus, at the point A = 0, Q is smaller,
T
Q
"
:
T
the smaller the value of a.
Further, it can be demonstrated that a
smaller value of a corresponds to a smaller value of Q in the earlier
T
part of its monotonic increasing region
(!..~.,
for small values of A),
22
but a larger value of
nT
from the point of intersection onwards.
that this observation also underlines the fact that any two
nT
Note
curves
with two different values of a intersect each other once and only once.
An intuitive understanding of this phenomenon is obtained from the fact
that Ql' which is the expected weight attached to Sl in the TE estimator,
2
is a monotonic decreasing function of a.
Hence, at every fixed value of
A, the smaller the value of a, the larger the weight attached to Sl on
the average, and the steeper the
tonic increasing region.
as
,"
:
nT
approaches
nb
nT
curve would tend to be in its mono-
However, the difference gradually tapers off
for very large values of A.
1
23
CHAPTER III
CONTINUOUS WEIGHT FUNCTION ESTIMATOR
Introductory Discussion
In Chapter I, it was indicated that a continuous function of the
preliminary test-statistic may be used as the weight in the convex
combination composed of the simple estimators of Sl.
If this function
is appropriately chosen, so that larger is the weight attached to b ,
l
say, the greater the chance (indicated by the observed value of the test
statistic) that it would be a "better,,9 representative of the parameter
(Sl) , then such a procedure can conceivably be an improvement over the
TE procedure of the previous chapter.
Note that in the latter (TE)
procedure, the weight attached to an estimator in the convex combination
for a given sample is either zero or one, and the "propensity" with
which it would be, say, zero rather than one over repeated sampling, is
controlled entirely by the level of significance (a) chosen for the
preliminary test.
On
the other hand, in the continuous generalization
proposed in this chapter, the weight function ranges for a given sample
over the entire spectrum of [0,1] interval, including the extreme points,
so that the TE estimator is just a special case of the Huntsberger (1955)
CWF estimator.
Moreover, the degree to which, say, Sl is represented
more prominently relative to b
.-
l
in the latter, is not dependent upon the
significance level a, but on the form of the weight function chosen •
Ideally, one would like to choose the weights in such a way that the MSE
:
of the linear (convex) combination becomes uniformly smaller than the
9"Better" here is in the sense of minimizing the MSE.
in Chapter II.
See footnote 2
24
of both, or at least of one of the extreme point estimators.
It is our
purpose in this chapter to study how this objective may be achieved, if
at all possible, and more generally, to investigate whether the continuous weighting procedure exercises a greater degree of control on the MSE
than does the TE estimator.
Definition of the Class of CWF
Estimators and Its Moments
The general form of the Huntsberger (1955) estimator of Sl is given
by:
(45)
where
~
u
is an appropriately chosen member, from the class C of all
continuous functions of the preliminary test-statistic u, satisfying the
following two conditions:
o -<
(i)
(ii)
~(O)
= 0;
~
u
~
< 1
u-
is a strictly monotonic increasing function of u, with
~(oo)
= 1.
The latter restriction is imposed so that a greater weight is
attached to b
l
in (45), the greater the chance that the null hypothesis
may be false.
In the light of (7), we can rewrite (45) as follows:
a
Z
:
=
Sl -
~ ~
all
(46)
b
u 2
Noting that Sl is independent of both u (and therefore, of
b , we can immediately write down the following:
2
~u)
and
25
E(Z)
(47)
Var(Z)
(48)
Hence, the first two moments of Z about Sl are given as follows:
(50)
Note that (47) and (49) are identical with (20) and (22) except that
ou
is now replaced by
~.
u
But unlike the TE estimator, the distribution
of Z does not involve the parameter a, whatever be the particular choice
of
~
within the class defined above.
u
Selection of a Weight Function Based
on an Optimality Criterion
One possible optimality criterion for the choice of a particular
member from the admissible class C of
function
~
* exists
u
then
:
~
* may
u
u
is unbiasedness.
Thus, if a
such that
E[Z *]
."
~
= Sl
for all values of A,
be called an unbiased weight function, and would thus
constitute a reasonable choice.
However, the following existence theorem
rules out this possibility.
Theorem 1:
Among the class of admissible weight functions, the only
unbiased weight function is
~
u
= 1.
26
The proof is immediate from (47), coupled with the fact that
S2 iff
~
=
u
1, since 0 -<
~
<
u-
1.
Still another optimality criterion for the choice of
uniformly minimum MSE about 6 ,
1
~
such that QZ*(A)
Thus, if
~
* is
~
u
would be
a member of the class C
u
QZ(A) for all values of A and for every other admissi-
ble weight function, with the inequality holding strictly for at least
~
one value of Aand one
function.
u
~
, then
* may
u
be called a minimum MSE weighting
However, the following theorem shows the nonexistence of such
a function.
Theorem 2:
Among the class of admissible weight functions, there
exists no function
~
*
u
such that
for every
Proof:
~
u
€
C and every A.
It is enough to show that for every (nontrivial) member
and QA for some A = A0' say, and
61
1
10
that Q is a continuous function of A in the neighborhood of A
It
~
u
€
C,
Q lies strictly between Q
Z
b
Z
0
.
will be easiest to show this for A = 0, (and, equivalently, for Sz
whenever 0
< p2 <
=0
1) since, at this point,
QA = Q (1-p2 )
b
61
1
,
a
.'
Q = Q (1-p2) + p2 ...12 E(~2b2)
b
Z
an
u 2
1
and 0 < E(~2b2) <
u 2
(j
2c
22
unless
<I>
u -
o or
1.
lOThis requirement is sufficient for the proof because b l and Sl are
the special (extreme) cases of a general CWF estimator, thus falling
within the admissible class C, and because neither of them is uniformly
"better" than the other over the entire range of A.
27
To show continuity of Q curve as a function of A, it is sufficient
Z
2 2
to show that each of the two terms in QZ' viz., E(~ub2) and 2S2E(~ub2) is
continuous in A.
This, however, is clearly true, since both the expecta-
tions are finite and can be written as linear functions of products of
two moments, each of which is a continuous function of A.
Q
z
Thus, indeed,
is a continuous function over the entire range of A.
Alternately, one can also consider some weaker optimality criterion
other than the one of uniformly minimum MSE, like minimizing the maximum
values of
~z
over A
(!.~.,
corresponding to a minimax choice of
~u),
or
maximizing the average efficiency of Z relative to the unbiased estimator
b
l
over the entire range of A, which can be represented by the integral:
00
J
[Qb (A) - QZ(A)] dA
I
o
However, both these criteria suggest the use of b , thus resulting in
l
a trivial solution.
Hence in the next section we will pursue a different
approach for selecting a weight function, appealing more to intuitive
reasoning than to any precise optimality considerations.
An Optimal Parametric Weight Function
Suppose the true value of the noncentrality parameter A were known.
Then, obviously, one would like to consider the parametric family of
weight functions;
!.~.,
the family identical to the class C, but whose
.0
members are functions of the parameter A, say t •
A
ll
The typical form of
:
11 The two extreme members of this class, viz., b and Sl, would
l
correspond to t A = 1 and t A = 0, respectively. Either of these would
amount to ignoring the knowledge of the actual value of A, except perhaps
in an ordinal sense. Thus'lfor example, if the true value of A were
known to be AO' where AO ~ 2' then we might choose the estimator Sl' or
28
the CWF estimator based on such a parametric weight function then becomes
A
Z£
= £A
(51)
b l + (l-£A)Sl
with
a
Bias (Zn)
~
MSE(Z£)
12
= ---
= nb
all
(52)
S (1-£,)
2
1\
[(1-p2) + p2£~ + 2p 2A(1-£A)2]
(53)
1
Since, for a fixed p and, by our assumption, known A, MSE(Z£) is a
differentiable function of £A alone, we can find £A* which minimizes the
MSE for each known value of A.
This is obtained as a solution to the
following equation:
so that the optimal parametric weight function for b
l
becomes
2A
1+2 A
(54)
and the resulting minimum MSE of the linear combination in (51) is
(55)
Thus, for example, if A = 0, then R. * = 0, and the optimal CWF
A
.-
estimator of Sl would be just Sl·
For A = ~
,
which is the critical
value of the preliminary test discussed in Chapter II, the optimal
estimator is obtained by giving equal weight to both b
l
and Sl in the
equivalently, £A = 0. On the other hand, the optimal choice of £A' as
indicated by (54) above, would involve a cardinal use of the knowledge
about A. Herein lies the inherent inefficiency of the TE procedure visa-vis the CWF estimation, the central theme of this thesis.
29
linear combination.
~
b
_!2
(1
1
P
2) 12
•
The corresponding value of the (minimum) MSE is
Finally, as A approaches 00,
*
~A
approaches one, and
MSE(Z~*) approaches ~b •
1
In the absence of specific information on A, then, our problem boils
down to one of finding a "reasonable" estimator of
*
~A'
Since A is the
parameter in the distribution of u, it is logical to restrict the choice
of such an estimator to functions of u.
It is in the context of approxi-
mating the optimal (in the sense of minimum MSE) linear combination
represented by
ou
that both the TE procedure through its weight function
Z~*
and the CWF procedure through its weight function
each other for a better control on the MSE.
ence between 0 and
u
~
u
~
u
must compete with
The only fundamental differ-
is the way each maps into the sample space (or,
equivalently, the u-space).
While 0
u
constitutes partitioning of the
sample space into acceptance and rejection regions of the UMP preliminary
test,
~
u
represents one-to-one mapping of the sample space.
The essential
question being asked here, then, is which is the more desirable mapping
for approximating
Z~*.
Some "Reasonable" Estimators of
*
~A
It can be shown that the uniformly minimum variance unbiased estimator
of 2A is given by
_0
12
Incidentally, this represents the point of sharpest departure between
the TE procedure, which would imply ~A = 0 and the CWF procedure, which prescribesA~A = 1/2, at A = 1/2.
Hence the gain in MSE(Z~*) relative to MSE(b1)
or MSE(Sl) should be the maximum at this point. It is for this reason that
A = 1/2 represents a reasonable choice for "normalizing" the weight functions
of the rival procedures with respect to their expected va1ueso However, note
that some other point, such as A = 0 may also constitute a reasonable choice
for normalization on different grounds.
30
Substituting this estimator in
*
~A
leads to the following expression
for q> :
u
\)
q>u = 1 - (\)-2)
1
u
However, this is not an admissible member of the Class C, since it
\)
violates condition (i) for u < \)-2
Another natural choice is to substitute 2A simply by the consistent
estimator u, whence the following consistent estimator of
* is
~A
obtained:
u
(56)
1+u
Although this constitutes an admissible weight function, it does not
facilitate evaluation of the expectations involved in (47) and (50). The
basic difficulty is centered around mathematical intractability of the
joint distribution of b
2
and u, which precludes derivation analytical
expressions for the bias and MSE of Z, whether in closed form or otherwise.
In the next section we will consider the family of exponential weight
functions, a case in which the aforementioned difficulty is resolved at
least sufficiently to allow evaluation of the required moments by
numerical integration methods.
..
The Family of Exponential Weight Functions
A typical member of this family is defined by the following:
:
(57)
13 A proof of this result (in the general case) is given by Toro (1968).
31
This family satisfies both the conditions imposed on the class C.
The corresponding Z estimator is given by:
(58)
From (57) the following results, necessary for the evaluation of the
desired moments, are easily verified:
Hence, after some algebraic manipulation, the bias and MSE of the
corresponding Z estimator can be written as follows:
(59)
MSE(Z )
e
..
(60)
32
It will be shown in the last section of this chapter that the three
expectations appearing in (59) and (60) reduce to the following integral
forms:
\)+1
2
(l-t)
1
2
exp{At-~\)
o
t
--l--}dt]
-t
(61)
\)
where a
o
=
(~\))2 [r(~)]-l
2
(62)
1
a2c
22
1
[j (l~t) 2 (l-t) 2 exp{At-2~\) l~t}dt]
e-A al
[j
\)+1
o
1
\)+3
1
-2-"2
t
(l"-t)
(l-t) exp{At-2~\) l_t}dt]
(63)
t
o
\)
where a l = -2-
2 a
o
Some General Comments on Bias and MSE of Z
e
(i) For A = 0 (which is equivalent to S2 = 0 for each fixed
p2
+1),
Ze is simply the convex combination of the two unbiased estimators b
l
and Sl' and hence obviously must itself be unbiased, which is verified
immediately from (59) and (61).
.-
Note that the same result is also true
for the TE estimator, and would indeed hold for any general
member of C.
~
u
which is a
14
l4 The condition of unbiasedness would constitute a reasonable criterion
for "normalization" of the weight function and thus would lead to the choice
of A = 0 as the point of normalization. (See the discussion in the section
on normalization of weight functions.)
33
Bias(Z )
= __-..,-e~ = _1_
Relative Bias
(ii)
6
+0 and every p2
Lemma:
For every 6
Proof:
Employing the transformation--1- t
2
> 0, R.B. < 1.
t
1
-6- F 1 = e
-A
ao
2
<
e
f
f
-A
o
00
u
(l+u)
3
2
-
u
2
F
1
v-2
2
e
=u
in (61), we can write:
-~vu
A u
e (l+u) du
(64)
00
v dx
e A Ga(x;Z)
=1
o
as required.
v
Ga(x;Z) above stands for the density function of Gamma
v
(r.v.) with parameter Z.
This inequality, of course, merely confirms
what would be clearly expected from the definition of Z.
e
A somewhat
stronger and more meaningful inequality is obtained by recognizing that
the upper limit on the bias of Z is actually attained when
e
+O.
when A = 0 but 6
2
00
f
p2
= 1,
i.~.,
Thus, from (64) above,
3
-
x
2
v
(X+~) Ga (x;Z) dx <
o
E(~)
2
d
Observing that ~ is a concave function (L~., dl [~] < 0),
we can apply Jensen's inequality
E(---L) < E(y)
~v+y
15
~v+E(y)
to obtain
1
= 1+2~
15This inequality stays if g(x) is a concave (convex) function of
the random variable x with finite expectation, then E[g(x)] ~(~ g[~x]
with equality iff pr[x = ~x] = 1. For a proof of this inequality see
Rao (1965).
34
~ £
Thus, for every value of
(0,00), we obtain the following strict
inequality on the relative bias:
•
I
(65)
R.B. < 1+2~
It is evident from the above derivation that this is a rather weak
inequality.
In fact, (64) can be used to express the exact result for
R.B. in the form of an expectation, quite similar to the corresponding
result for the TE estimator as given by (34):
R.B. =
I
=
~~
2
3
A
e- E{[g(x)]2 exp[Ag(X)]}
(66)
where
x
g(x) =rl~v'
(iii) Lemma:
v
x ~ Ga(Z)
For every fixed value of A and v, the R.B. is a
monotone decreasing function of
Proof:
(67)
~.
Note that intuitively this is immediately obvious from (57),
since a higher value of
~
corresponds, one-to-one, to a larger weight
for the unbiased estimator b
in the linear combination comprising Ze.
l
A formal proof follows from the observation that the integrand in (64)
is a monotonic function in
~
(although interestingly enough, this is not
obvious from the result (61».
(iv) For every fixed value of
~
and v, the R.B. is a monotone
function of A, and approaches zero as A approaches infinity.
This follows at once from the fact that the integrand in (61) is
a monotone decreasing function of A over the entire range of integration,
and indeed, approaches zero as A + 00.
However, as in the case of the TE
estimator, the absolute value of the bias is not a monotone function of A.
35
Perhaps it is of some interest to observe that the above conclusions
on the bias of Z identically follow the general pattern in the case of
e
the TE estimator.
The difference between the two estimation procedures
is one of degree of control, not of mutually opposite trends of behavior.
This will apply equally to a comparison of the MSEs.
Comments on MSE
Going through the same transformations employed in obtaining the
results (64), (66) and (67), we can express (6Z) and (63) in terms of
the following expectations:
3
F
Z
2
= E(b 2
e
_eu
~
=0
)
2
cZZe
-A
E{[g(x)]
2
exp[Ag(X)]
(68)
5
+ ZA [g(x)]Z exp[Ag(X)]}
3
= 02cZZe-A
E{[h(x)]Z exp[Ah(x)]
5
(69)
+ ZA [h(x)]2 exp[Ah(x)]}
where
x
h (x) = -:-':::'-:--
(70)
X+Z~v
Hence, using the result (60), we can establish the following:
MSE(Z ) =
(71)
e
..
where
H(A)
=
=
2
F1
[F 3 + 4AO c zz Sz
ZF Z]
3 5 3
A
e- E{ZA[eAg(gZ- gZ)] _ (l-ZA)eAg gZ
2
(0 c ZZ )
-1
5
5
-3
-3
Ag
Ag
Ah
Ah
Z
_ [e
gZ _ e
h ] _ ZA[e
gZ _ e
hZ]}
(72)
36
Thus, from (71),
<
Ze > Qb1
Q
•
Since 0 < hex) < g(x)< 1 V
parentheses {
according as H(A)
<
0
>
X E
(0,00) each bracketed term ([
]) in the
} of (71) is non-negative.
Note the similarity of the expressions in (71) and (72) with the
corresponding result (41) for the MSE of the TE estimator.
However, it
should be pointed out that since the expectations of products of functions
of g(x) and hex) involved in (72) are not mathematically as tractable as
the Q functions of Chapter II, we will have to rely considerably on the
numerical results obtained in the next chapter for drawing conclusions about
the MSE function given by (71) and (72).
Nevertheless, the following gen-
era1izations are possible on analytical grounds:
(i)
-23
-23
3
2
H(A=O) = E{- g (x) - [g (x) - h (x)]
=
3
2
E{-g (x)
(73)
3
t;v
2
[2 - (1 - x+2t;) J} < 0
Further, since the g function is monotonic decreasing in t;
(throughout the range of x, and for each given v), it can be shown from
(73) that H(O) is monotonic increasing in t;.
(~.~.,
Thus, at the point A = 0,
for S2 = 0), Q is smaller, the smaller the value of
Z
~.
Moreover,
e
."
since 0 < g(x) < 1, we can write
/H(o)1 < E{g(x) [1 + 1 - (1 - x+2t;v
)2]}
t;v
t;v
t;v
= E{g(x) [1 + (x+2t;v ) + (1 - x+2t;v ) (x+2t;v )]}
t;v
- E{g(x) + hex) (x+2t;v) + hex) (1 - g(x»}
37
<
E{g(x) + Zh(x) (1 - g(x)}
<
E{g(x) + Zh(x)}
Since g(x) and h(x) are both concave functions of x, we can apply
Jensen's inequality (see footnote 15) to obtain
1
IH(O)I <
=
Z
l+Z~ + 1+4~
4
1
1+4~
(1+Z0 (1+40
4
< -1+4~
Thus, a lower bound on
nZ
relative to
e
nb
at A = 0 is given by
1
(74)
(ii) To study the behavior of the MSE function with respect to A, we
rewrite (7Z) as follows:
H(A) = E -Z(l-ZA) e
-A(l-g)
1.Z
g
5
~
-A(l-h) 2
-A(l-g) Z - e
h ]
- 4A[e
g
-3
-5
A
Z
+ e- (l-h) [h _ 2Ah 2 ]
where g and h denote functions of x defined previously.
After some
algebraic manipulation, it can be shown that,
H' (A)
=
5
3
A
2
E{Zh e- (l-h) [l-A(l-h)] + Z(1_g)g2 e- A(l-g)
3 3 5
_ h2 (l-h) e- A(l-h) + 4[1_A(1_g)]e- A(1- g )(g2 _ g2)}
38
Whence, using the inequalities 0 <
1
28
< h < g < 1, we can establish the
following:
5
•
HI (A)
>
3
5
E{2h2 e-A(l-h) (I-A) + 4(1_A)e- A(1-g)(g2 _ gZ)
3
+ h2 e-A(l-h) [I-g) - (g-h)]
> 0 whenever A > 1
Thus, for all values of
and v, the maximum of H(A) occurs to the right
~
of A = 1, and consequently
nz
e
is monotonic increasing in A in the interval
~
[0, l+w] , where w>O is determined by the particular choice of
and v.
(iii) Next, let us study the behavior of H(A) at the critical value
1
A=Z:
1
-;::;-g
2
=e
2[e
5
1
5
31 h 2 ]}
g2 _ e 2
-5
-
-5
(g2 _ h2)]}
(75)
5
5
(g2 _ h2)]}
Unfortunately, it does not seem possible to extend this result in a
way which would indicate definitely if H(f) < 0 for at least one value
of~.
However, it will be observed from the numerical results of the
next chapter that this result is, in fact, true for several different
.-
choices
of~.
Indeed, it is this particular observation that brings
into a sharp focus the difference between the TE procedure of the
previous chapter and the CWF procedure of this chapter.
Normalization of Weight Functions
It will be recognized that both 0 , which corresponds to the TE
u
weighting procedure, and ~e corresponding to the exponential weighting
u
39
procedure represent a family of weight functions--the former defined over
the range of the parameter a, and the latter over the range
of~.
In
both cases, each member of the respective family will display a distinct
pattern of bias and MSE of the corresponding estimator of Slover the
range of the noncentrality A.
The question may naturally arise, then,
whether it would be possible to normalize the two weighting procedures
in some sense before comparing their effectiveness in controlling the
bias and MSE of the resulting estimator of Sl.
It was pointed out
earlier that the essential difference between the two weighting procedures lies in the way 0
u
and
~
u
map into the u space, or concurrently,
the expectations of the weights map into the parameter space
A space).
(i.~.,
the
Hence one logical criterion for normalization of the two
weighting procedures would be to equalize the expected weights at A
1
2·
This value of A is preferred because it is the critical value of the MSE
test
(i.~.,
it forms the boundary between the corresponding null and
alternative hypotheses), and also because, as evident from the preceding
discussion, the neighborhood around A =
t
is crucial to the behavior
pattern of the MSE curves of the TE and CWF estimators.
(See, also,
footnotes 12 and 14 in the preceding discussion.)
An incidental advantage of choosing A
= 21
for normalization of the
weight functions is that the expectation of 0u at A =
size of the preliminary test, which is a.
very definition, 0
u
t
is merely the
This is so because by its
is just the test function corresponding to the
normalized UMP test, so that E(o u ) = S(A) is the power function of this
16
test.
Moreover, by virtue of the well-known property of a one-tailed
l6 S (A) may be interpreted as the probability of rejecting the null
hypothesis when the true parameter value is A.
40
liMP unbiased test, e(A) is a monotonic increasing function of A with
e(f)
= a and e(oo) = 1 for each a
(0,1).
E
We will also observe a little
later that for each fixed value of A (and v), e(A) is larger, the greater
the value of a,
!.~.,
if a
'"
> a , the curve e (A) corresponding to a
"
entirely above the curve e " (A) corresponding to a.
,
lies
It is worth pointing
out that both these properties of the power function are basic to an
understanding of the behavior pattern of the bias and MSE of the TE
estimator for different values of a, and over the range of A.
(See the
discussion on bias and MSE of T in Chapter II.)
The expected value of the exponential weight function, on the other
hand, does not lend itself to such a meaningful interpretation.
It can,
however, be evaluated numerically precisely in the same manner as other
expectations involved in the moments of Z.
e
As derived in the next
section, it is given by the following integral form:
where
=e
-A
1
a
o
f
t
(l-t)
v-1
2
(l-t)
3
2
o
t
expO.t-E;v -l-}dt
-t
(76)
Again, using a similar analysis as adopted in reaching the conclusions
on the relative bias (see pages 32 and 33), one can easily verify the
following conclusions:
o
(i)
< F
o
so that 0 < E(~e) < 1
u
< 1,
In fact, it is easily seen from (64), (66) and (67) that
1
F
o
1
2
= E{e
-A[l-g(x)]
2
g (x)}
1
<
2
E{g
since g (x) is a concave function of x.
[E(x)]
(77)
41
Thus,
(78)
(ii) F is a monotonic decreasing function of A,
o
and approaches a as A approaches infinity.
~
and v being fixed,
(Correspondingly,
E(~
e
u
) is a
monotonic increasing function of A.)
(iii) For each fixed value of A and v, F
o
~.
is monotonic decreasing in
Hence E(~e) is higher, the greater the value of ~.
u
From the result (77) and the preceding discussion, it follows that
1
normalization of the two weighting procedures at A = 2 boils down to the
problem of solving, corresponding to every choice of a, the following
equation for
~:
111
= e-
1 - a
On
-
~2 (x)
-
2 E[g2(x) e
]
the other hand, normalization at A
following equation for
I
1
w
(79)
=a
requires solution of the
~:
v
(2' '2)
=
1
2
E[g (x)]
a
i.~.
,
ex>
J
o
.-
x
(x+~)
1
2
-
v
Ga(x;Z) dx
(80)
It will be seen from numerical results in the next chapter that the
equations in (79) and (80) do not, in general, yield the same solution
for
~.
In his investigation on TE procedures in a factorial model, Gun (1965)
came to the conclusion that a value of a very close to .50 was nearoptimal in the context of mean squared error loss functions.
In the
42
=
framework of the present discussion, it is seen that a
locally unbiased weight function at A
*
tA_l.
1
= 2'
.50 provides a
since for this a, E(ou)
=
Intuitively, this is equivalent to an assurance that on an
2
average one would be perfectly impartial between accepting or rejecting
the hypothesis, whenever the true value of the parameter lies on the
boundary of the null and alternative hypotheses, and that, as it turns
out, is a most desirable thing to do in the context of the TE procedure.
It will be seen from the numerical results in Tables land 2 that the
choice of a
A=
f'
= .5
and to
corresponds, roughly, to
~
= .5
for normalization at
~ = 1 for normalization at A = O.
Derivation of Expectations Occurring in the Moments of Z
e
F = E(b2e-~u)
l
(i)
00
=
J J
o
b
00
_00
b
2
exp{-~
2
2
2
s c
2
2
2(~)
0'2
)
d b ds
}N(S2 '
c22 Xv 0'2
2
22
where N(~,n2) represents the normal density with parameters ~ and n 2 ,
and X2 (y) represents the central X2 density with the random variable
v
y and parameter v.
Let
00
00
y exp{- -~ y 2
2
-00
k
f
.'
=
1
1
2
(ZTIn 2)
00
Jy
exp{-
-00
where
~
=
S2'
n2
= 0' 2 c 22 ,
k
Z
=
2
s c
e2
[y
Z
22
,
2
- ~+~]}
n 2e2
n 2 e2
e2 = ~+.L2
k2
n
'
d y
e
Table 1.
~
~
.•
e
*a
Values of E(ou) for selected values of A, a and v compared with £A
0
.2500
.5000
.3333
.026
.013
.010
.038
.031
.028
.042
.037
.035
.050
.050
.050
.053
.031
.026
.077
.064
.061
.084
.076
.074
.100
.100
.100
.338
.305
.299
2
8
18
.423
.347
.331
£A*
0
.425
.410
.407
.511
.451
.538
.482
__ .~3~ ___ .~7~ _
.333
.500
.500
.500
.451
.442
.440
.400
1
.586
.540
.500
7.564
9.000
.131
.231
.264
.205
.432
.504
.291
.650
.741
.336
.746
.833
.383
.825
.900
.248
.369
.404
.372
.603
.660
.501
.803
.857
.564
.874
.919
.624
.925
.958
.813
.84a
.854
.930
.956
.960
.980
.991
.993
.990
.997
.997
.996
.999
.999
.871
.870
.960
.964
.991
.994
.996
.998
.998
.999
~6~3 ___ ~8~O___ ~9~6___ ~9~4___ .~9~
__ .999
= .05
= .10
= .50
.622
.642
.646
u
_.~3~
6.250
.145
.175
.184
a
2
8
18
4.000
.074
.096
.102
a
2
8
18
2.250
1.000
a
2
8
18
e
_
a
= 1 (a
1
_
.704
.678
.667
~ .50)
.818
.889
.926
.938
aThese values have been reproduced from the relevant tables (Tables 3, 4, 6, 12) in Toro's
dissertation, ~. cit.
.947
~
w
e
Table 2.
••
Values of
~
0
2
8
18
.292
.210
.195
-
.
e
E(~u)
.2500
for selected values of A,
.3333
.5000
~
e
and v compared with £A*
1.0000
2.2500
~
.358
.277
.262
.378
.298
.282
.417
.338
.322
.515
.444
.428
.397
.320
.305
.470
.402
.388
.492
.427
.413
.534
.473
.460
.636
.590
.580
.452
.381
.368
.527
.465
.454
.549
.491
.480
.590
.538
.528
.511
.446
.436
.584
.531
.523
.606
.556
.549
.646
.602
.596
.741
.714
.710
=
.810
.793
.789
.878
.895
.897
.906
.929
.931
.928
.952
.956
.902
.903
.903
.957
.965
.967
.972
.980
.982
.982
.989
.991
.932
.935
.936
.974
.980
.982
.984
.990
.991
.991
.995
.996
.954
.958
.959
.985
.989
.990
.992
.995
.996
.996
.998
.998
.999
.999+
1.0
.878
.873
.873
~
2
8
18
=
9.0000
.70
.840
.830
.828
.690
.654
.647
~
2
8
18
=
7.5625
.50
.796
.778
.774
~
2
8
18
=
6.2500
.25
.682
.636
.624
~
2
8
18
=
4.0000
2.0
.616
.572
.545
.694
.649
.627
.712
.671
.651
.746
.711
.694
.826
.805
.795
.930
.927
.924
.980
.981
.981
.995
.997
.997
.998
.999
.999+
0
.333
.400
.500
.667
.818
.926
.938
.947
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - £A*
.p.p-
45
After completing the square in the exponent of the integrand, we will
obtain the following:
1
1 I
~2
1
~
1
D1 = (n 28 2 ) exp {2n 2 (n 28 2 -1)} E(y), where yrv N(n 28 2 '8 2 )
1
1 2
~
~2
1
= (n282)
(n282) exp{ 2n 2 (n 28 2 - 1)}
=
->.. (
S2 e
Hence, F = S2 e
1
->..
k
2
k +2E;n 2
00
a
3
2
f
0
(
0
)2 exp
k
(
k
2
k +2E;n 2
3
2
2
k +2E;n 2
2
)2 exp{>..
) }
k2
2
k +2E;n 2
2
2 ~- 1
k
k
2
v 2n 2 } (2E;n z)
v
2
v
where a o = (E;v) /f(I)o
Now, making the transformation
k2
2
= t,17 we will obtain the
k +2E;n 2
following expression:
F
1
->..
= S2 e a 0
->..
= S2 e a 0
f
1
V
t ;l (l-t)
-v+2
2
t
exp{At-vE; 1-t } d t
0
1
f
v+1
t
2
(l-t)
(l-t)
1
2 exp{At-E;v t
1-t } d t
0
This is precisely the result (61)0
00
2
E(b 2e-E;u) = f D X2 (vs ) d s 2
2 \) ?
2
0
2
00
b
2 } N(S2,cr f c )d b
exp{-E;
where D = f b;
2
2
22
2
-00
s c
22
1
~2
1
2
1 2
= (n 82) eXP{2n 2
(n 28 2 '- I)} E(y )
2
(11)
,0
;
F
2
=
46
y~N(n2~2 ' ~2).
where, as in (i) above,
2
E(y )
= e1 2 (1
D
2
Substitution of the result
21..
+ n2 e2 ') leads to the following expression for D :
2
= e- A(
k
2
22 2
k
}(k n
)[1 + 2A( 2k
)]
k 2+2E,;n 2 k 2+2E,;n 2
k +2E,;n 2
2!
)2 exp{A
k2+2~n2
~
Hence, by direct comparison with the expressions for D and F above,
l
l
we can obtain the following:
d t
This coincides with the result in (62).
(iii)
F
3
=
E(b~e-2E,;U)
J
0
2
2
where D* = J b; exp{-2E,; 2
2
-00
s c
= e2
=
Z
2
d s
D2* Xv2 (~)
02
b
00
where y2
00
}N(S 2,02 c22 )d b
2
22
+ 2 E,; ,E,; ~ N(~ , \ ) •
kZ
n y
y
By comparing J * with J , and
2
2
working through analogous steps, we can arrive at the following:
1
=
F3
2 -A
neal
J
o
1
J
..
;
o
v
where a
17
l
=2
2
t
(-)
l-t
'1+1
2
(l-t)
1
2
t
exp{At-2E,;v l-t} d t
'1+3
1
t
(_t_) Z (1-t)2 exp{At-ZE,;v 1-t}
d t
1-t
a •
o
2
Note that use of the more "natural" transformation u
k
= 2E,;n
2
here
directly gives the result in (64), and the corresponding results for
other integrals. The only preference for the t-transformation lies in
the possible advantage of the resulting limits for numerical evaluation
of the definite integrals.
47
(iv)
2
00
=
f
Do
o
b2
00
where Do =
f
~
exp{-t;
s c
-00
X~ (V~2
) d S2
} N(S2' cr 2c 22 )d b 2
22
1
1 Z
~2
1
(n2e2) exp {2n 2 (n 2e 2 -1)}.
Hence,
e
-A
2
1.
k2
k2
k2 v - 1
k
)2
exp{A
}
(
)2
-v 2n2
2'"n2
2
o f (2
2
2
o k +2t;n
k +2t;n
~
00
a
2
k
x d (Zt;n 2 )
1
v-1
= e-A a f (_t_) 2 (1-t)
o
1-t
3
2 exp{At-t;v 1~t} d t
o
= Fo ,
;
say.
Thus, E(~e)
u
=1
- F •
0
48
CHAPTER IV
NUMERICAL RESULTS AND COMPARISONS
Prologue
In this chapter, we will examine some results on the bias and MSE of
Z for selected values of the various parameters involved.
e
Also, expec-
tat ions of the exponential weight function ~e over the range of the
u
noncentrality parameter A are evaluated to compare its "bias" in estimating the optimal parametric weight function
*
~A'
These results are then
compared with their counterparts for the TE estimator to draw some
definitive conclusions about the relative merits of the two estimation
procedures.
Since the results in the case of the TE estimator for selected values
of the relevant parameters were already obtained by Toro (1968) in his
dissertation, they will merely be reproduced
18
below in appropriate
contexts for a comparative study--no attempt has been made here to
duplicate the numerical calculations or to focus attention on a separate
analysis of these results, although a curious and agile reader will
find in them verification of all the analytical properties discussed in
Chapter II.
Choice of the Parameter Values
In order to utilize all the results obtained by Toro for a
comparative study of the TE and CWF procedures, the same parameter
;
values chosen by him were employed, as a beginning, also for
18 I am, indeed, grateful to Dr. Toro and to the Department of
Economics for permission to use these results.
49
calculations involving the Z estimator.
e
19
6 = .1, .4, l.0, 2.0, 4.0
2
p
=
n
= 5,
These values were as follows:
.2, .4, .6, .8
11, 21,
i.~.,
v
= 2,
8, 18
The resulting values of A covered a wide range of interesting cases.
20
However, since the analytical conclusions about the properties of
the Z estimator (as described in Chapter II) were less definitive than in
e
the case of the TE estimator, it was found desirable to choose additional
values of 6 and p, so as to include more values of A of particular
2
interest, and to study the impact of multicollinearity in greater detail.
These additional values, included for calculations on Z , were as follows:
e
6
=
p
= .5, .75, .9, .96
2
.8, 1.2, 1.5, 2.5
Table 3 below contains values of A for different combinations of 6 , v
2
and p.
Apart from the three parameters common to both the estimation
procedures, the TE estimator contains the parameter a which plays a
"manipulative" role in controlling its bias and MSE, as explained in
19
..
..
Subsequently, for reasons of space, results corresponding to 62=4
were omitted in the presentation of tables, since they almost uniformly
reflected asymptotic convergence of both bias and MSE, and were of no
other information value •
20Toro 's results are based on a slight simplification of the model
being considered here, as given by (1). He assumes, like Bancroft, that
all the variables in the model are standardized with respect to the
sample mean and standard deviation, and that E ~ N(O,l). This implies,
i
in the notation of Chapter II, that all = a 22 = n-1, a 12 = (n-1)p and
A = ~ 6~ (n-1) (1-p2).
With these changes, all the results obtained in
the previous two chapters are identical for the specified model, and
the numerical results in this chapter are based on this modification.
e
Table 3.
:s:J
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
e
'.
Values of the noncentra1ity parameter A for se1eeted combinations of S2' p and v
.20
0.0192
0.3072
1.2288
1.9200
2.7648
4.3200
7.6800
12.0000
:
.40
0.0168
0.2688
1.0752
1. 6800
2.4192
3.7800
6.7200
10.5000
I
.50
I
I
.60
v
0.0150
0.2400
0.9600
1.5000
2.1600
3.3750
6.0000
9.3750
0.0128
0.2048
0.8192
1.2800
1.8432
2.8800
5.1200
8.0000
.4
.8
1.0
1.2
1.5
2.0
2.5
0.0480
0.7680
3.0720
4.8000
6.9120
10.8000
19.2000
30.0000
0.0420
0.6720
2.6880
4.2000
6.0480
9.4500
16.8000
26.2500
0.0375
0.6000
2.4000
3.7500
5.4000
8.4375
15.0000
23.4375
0.0320
0.5120
2.0480
3.2000
4.6080
7.2000
12.8000
20.0000
.4
.8
1.0
1.2
1.5
2.0
2.5
0.0960
1.5360
6.1440
9.6000
13.8240
21.6000
38.4000
60.0000
0.0840
1. 3440
5.3760
8.4000
12.0960
18.9000
33.6000
52.5000
0.0750
1.2000
4.8000
7.5000
10.8000
16.8750
30.0000
46.8750
0.0640
1.0240
4.0960
6.4000
9.2160
14.4000
25.6000
40.0000
I
.80
I
.90
I
.96
=2
0.0072
0.1152
0.4608
0.7200
1.0468
1.6200
2.8800
4.5000
0.0038
0.0608
0.2432
0.3800
0.5472
0.8550
1.5200
2.3750
0.0016
0.0251
0.1004
0.1568
0.2258
0.3528
0.6272
0.9800
0.0180
0.2880
1.1520
1.8000
2.5920
4.0500
7.2000
11.2500
0.0095
0.1520
0.6080
0.9500
1. 3680
2.1375
3.8000
5.9375
0.0039
0.0627
0.2509
0.3920
0.5645
0.8820
1.5680
2.4500
0.0360
0.5760
2.3040
3.6000
5.1840
8.1000
14.4000
22.5000
0.0190
0.3040
1.2160
1.9000
2.7360
4.2750
7.6000
11.8750
0.0078
0.1254
0.5018
0.7840
1.1290
1.7640
3.1360
4.9000
=8
0.0219
0.3500
1.4000
2.1875
3.1500
4.9219
8.7500
13.6719
v
.1
.75
0.0088
0.1400
0.5600
0.8750
1. 2600
1.9688
3.5000
5.4688
v
.1
e
= 18
0.0438
0.7000
2.8000
4.3750
6.3000
9.8438
17.5000
27.3438
VI
0
51
Chapter II.
A similar function characterizes the parameter I; in the 2
estimator.
Toro considered three different significance levels in his
e
investigations:
(i)
a. = .05 corresponding to what he called the "mean square"
sequential (TE) estimator
(!.~.,
the estimator resulting from the MSE
preliminary test with the null hypothesis A
(ii) a.
estimator
=
~
21 ).
.05 corresponding to what he called the "usual" sequential
(!.~.,
the TE estimator resulting fr0m the standard ANOVA
preliminary test with the null hypothesis A = 0).
The corresponding
significance levels for the MSE test are approximately between .10 and
.15 for the three values of n being considered here.
(See, for example,
Table 4 in Toro (1968), p. 42.)
(iii) The value of a. corresponding to the critical value one of the
test-statistic (1. e., u = 1).
- a.
Toro called the resulting estimator "50
percent" sequential estimator, implying that it coincided approximately
with a Type 1 error of .5.
Actually, as indicated in Table 1, the values
of the power function at A =
21
(!.~.,
the size of the MSE test) were
.586, .540 and .531 for v = 2, 8 and 18, respectively.
In order to study the influence of the "manipulative" parameter I; on
the bias and MSE of 2 , we have selected five different values of 1;:
e
I; =
.25, .50, • 70, 1. 0 and 2.0.
Comparison of Relative Bias
It was already seen in the result (66) that the relative bias of 2 ,
e
Bias (2 )
!.~.,
the ratio
e
---""'A~
is (explicitly) a function of A, v and 1;.
Bias «(31)
Further, it was pointed out in the previous chapter that the relative
52
bias (i) decreases monotonically with A for every fixed
(iii) decreases monotonically with A for every fixed
\I
\I
and A; and
and~.
A study
of the numerical results presented in Tables 4, 5 and 6 below as well as
Figure 2 amply demonstrate both these conditions.
As
in the case of the expected weight function, the values of the
relative bias were obtained using the same subroutine for numerical
integration
21
(utilizing Simpson's rule).
The integral form used for
evaluation of the bias was the one in formula (61).
Since the relative
bias is a function of A independently of the particular values of
p
and
8 , the values in the following tables should be read alongside the A
2
values presented in Table 3.
fixed value of
~
and
over the range of A.
\I
Thus, we have 64 values available for each
to study the behavior pattern of relative bias
These results clearly bring out the inverse
relationship between A and the relative bias, with the latter approaching
zero as A approaches infinity.
Incidentally, this systematic behavior
over such a wide range of A values also provides an excellent check on
the accuracy of the numerical results.
The relative bias is highly sensitive to changes in the value of
As
an examination of Tables 4, 5 and 6 and of Figure 2 reveals, not only
is it monotonic decreasing in
as
~.
~
~,
but the rate of decrease gets smaller
increases.
The effect of
\I,
or the sample size, on the relative bias is somewhat
less simple, but still quite reasonable.
For a given
~,
the bias tends
~-
21
The Appendix lists the Main and Subroutine FORTRAN programs used for
all numerical calculations.
e
Table 4.
e
~.
1
Values of PS
E(I\-Ze) for selected values of 13 , P and C;, and \)
2
2
N
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.20
I
.40
I
.50
I
I
.60
C;
0.4250
0.3863
0.2906
0.2380
0.1888
0.1420
0.0740
0.0270
0.4250
0.3906
0.3044
0.2548
0.2073
0.1598
0.0880
0.0352
0.4250
0.3945
0.3150
0.2684
0.2225
0.1753
·0.1008
0.0434
0.4250
0.3988
0.3290
0.2863
0.2432
0.1832
0.1200
0.0701
.1
~4
.8
1.0
1.2
1.5
2.0
2.5
0.2850
0.2500
0.1700
0.1290
0.0933
0.0537
0.0185
0.0058
0.2850
0.2544
0.1809
0.1418
0.1063
0.0647
0.0248
0.0085
0.2850
0.2575
0.1898
0.1522
0.1173
0.0747
0.0309
0.0115
0.1675
0.1425
0.0862
0.0595
0.0388
0.0183
0.0042
0.0008
0.1675
0.1456
0.0934
0.0678
0.0462
0.0235
0.0062
0.0014
0.1680
0.1480
0.0995
0.0744
0.0527
0.0287
0.0085
0.0022
0.1683
0.1508
0.1075
0.0837
0.0621
0.0366
0.0126
0.0037
I
.80
=2
I
.90
I
.96
= .25
0.4250
0.4106
0.3677
0.3391
0.3078
0.2593
0.1832
0.1366
0.4256
0.4181
0.3940
0.3771
0.3578
0.3253
0.2669
0.2098
0.4260
0.4229
0.4126
0.4051
0.3962
0.3804
0.3490
0.3132
0.2850
b.2722
0.2342
0.2098
0.1839
0.1451
0.0894
0.0504
0.2856
0.2786
0.2574
0.2426
0.2257
0.1982
0.1510
0.1081
0.2865
0.2831
0.2738
0.2672
0.2593
0.2454
0.2182
0.1882
0.1688
0.1588
0.1309
0.1134
0.0954
0.0698
0.0366
0.0168
0.1689
0.1639
0.1478
0.1369
0.1247
0.1053
0.0737
0.0472
0.1698
0.1672
0.1602
0.1551
0.1492
0.1390
0.1194
0.0984
= .50
0.2850
0.2690
0.2245
0.1965
0.1677
0.1264
0.07;4
0.0366
0.2850
0.2617
0.2012
0.1663
0.1328
0.0894
0.0410
0.0169
C;
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.70
0.4250
0.4073
0.3563
0.3233
0.2880
0.2348
0.1703
0.1118
C;
e
,'
= 1.0
0.1687
0.1567
0.1238
0.1041
0.0846
0.0582
0.0269
0.0108
lJ1
w
e
Table 5.
.
e
\.
1
Values of pS
e
'
E(Sl-Ze) for se1eeted values of S2' p and
~,
and v
I
.80
=8
2
:sJ
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.20
I
.40
I
.50
I
I
.60
~
0.5000
0.3925
0.1888
0.1120
0.0613
0.0217
0.0030
0.0004
0.5000
0.4050
0.2122
0.1340
0.0781
0.0307
0.0050
0.0007
0.5000
0.4150
0.2323
0.1534
0.0942
0.0403
0.0077
0.0012
0.5017
0.4275
0.2596
0.1813
0.1186
0.0564
0.0131
0.0025
0.3250
0.2300
0.0794
0.0370
0.0154
0.0033
0.0002
0.0000
0.3250
0.2400
0.0944
0.0480
0.0219
0.0055
0.0004
0.0000
0.3260
0.2486
0.1075
0.0586
0.0287
0.0083
0.0007
0.0001
0.3267
0.2592
0.1263
0.0748
0.0403
0.0136
0.0016
0.0001
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
0.1800
0.1138
0.0275
0.0100
0.0029
0.0003
0.0000
0.0000
0.1800
0.1212
0.0350
0.0140
0.0048
0.0007
0.0000
0.0000
0.1800
0.1270
0.0418
0.0184
0.0070
0.0012
0.0000
0.0000
0.1817
0.1342
0.0517
0.0257
0.0111
0.0024
0.0001
0.0000
.90
I
.96
0.5038
0.4606
0.3463
0.2810
0.2188
0.1401
0.0565
0.0193
0.5056
0.4819
0.4140
0.3699
0.3228
0.2424
0.1510
0.0806
0.5063
0.4966
0.4663
0.4448
0.4200
0.3782
0.3026
0.2287
0.3288
0.2884
0.1914
0.1415
0.0984
0.0513
0.0135
0.0028
0.3300
0.3081
0.2476
0.2106
0.1730
0.1211
0.0573
0.0228
0.3312
0.3216
0.2938
0.2745
0.2529
0.2174
0.1576
0.1050
0.1838
0.1544
0.0898
0.0601
0.0371
0.0154
0.0025
0.0002
0.1844
0.1683
0.1262
0.1019
0.0786
0.0489
0.0179
0.0051
0.1844
0.1781
0.1581
0.1447
0.1298
0.1064
0.0694
0.0404
= .50
0.3280
0.2800
0.1703
0.1184
0.0766
0.0352
0.0073
0.0011
~
I
= .25
0.5033
0.4510
0.3195
0.2484
0.1841
0.1082
0.0370
0.0106
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.75
= 1.0
0.1827
0.1487
0.0770
0.0475
0.0264
0.0092
0.0011
0.0000
lJ1
~
e
Table 6.
e
~
1
Values of pS
e
••
E(Sl-Ze) for selected values of S2' p and
~,
and v
I
.80
= 18
2
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.20
I
.40
I
.50
I
I
.60
~
0.5100
0.3163
0.0731
0.0260
0.0075
0~0010
0~0000
0.0000
0.5125
0.3369
0.0928
0.0370
0.0125
0.0020
0.0000
0.0000
0.5130
0.3530
0.1110
0.4840
0.0-182
0.0093
0.00&1
0.0000
0.5150
0.3746
0.1385
0.0677
0.02M
0.0066
0.0003
0.0000
0.3250
0.1625
0.0188
0.0040
0.0008
0.0000
0.0000
0.0000
0.3300
0.1781
0.0266
0.0068
0.0012
0.0000
0.0000
0.0000
0.3300
0.1905
0.0345
0.0100
0.0023
0.0001
0.0000
0.0000
0.1750
0.0688
0.0038
0.0005
0;0000
0.0000
0;0000
0.0000
0.1775
0.0781
0.0059
0.0010
0.0002
0".0000
0.0000
0.0000
0.1780
0.0"860
0.0088
0.0016
0.0002
·0;Qt)00
0;0000
0".0000
0.1800
0.0962
0.0135
0.0032
0.0006
0.0000
0.0000
0.0000
.90
I
.96
0.5200
0.4347
0.2463
0.1623
0.0984
0.0403
0.0065
0.0008
0.5222
0.4761
0.3515
0.2808
0.2140
0.1310
0.0469
0.0133
0.5240
0.5055
0.4456
0.4056
0.3618
0.2936
0.1882
0.1076
0.3375
0.2584
0.1120
0.0605
0.0289
O. 0077
0.0005
0.0000
0.3400
0.2953
0.1893
0.1360
0.0911
0.0441
0.0096
0.0015
0.3417
0.3224
0.2681
0.2335
0.1974
0.1452
0.0754
0.0329
0.1825
0.1288
0.0422
0.0185
0.0068
0.0011
0.0000
0.0000
0.1844
0.1536
0.0850
0.0547
0.0320
0.0121
0.0015
0.0001
0.1865
0.1724
0.1350
0.1124
0.0899
0.0597
0.0248
0.0082
.50
0.3360
0.2433
0.0883
0.0420
0.0173
0.0036
0.0001
0.0000
0.3317
0.2079
0.0479
0.0165
0.0047
0.0006
0.0000
0.0000
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
=
I
.25
0.5187
0.4170
0.2097
0.1269
0.0697
0.0240
0.0028
0.0002
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
=
.75
= 1.0
0.1813
0.1187
0.0308
0.0113
0.0034
0.0004
0.0000
0.0000
V1
V1
56
ll"l
N
.
0
ll"l
.
II
II
W'
W'
,
/
/
/
/
,,,
/
/.
.
<Il
ll"l
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57
to become higher as v increases, for very small values of A.
But the
rate at which it decreases with increasing A is higher, the higher the
•
value of v, so that on the whole, there is a definite advantage in having
a larger sample size.
The results for the bias of the TE estimator were obtained by Toro
(1968) and presented in Table 14 of his dissertation.
Here we are pri-
mari1y interested in studying the relative behavior of the two bias
functions.
In Table 7 are presented the values of the bias of T relative
to the bias of 6 , so as to make them comparable with Tables 4, 5 and 6
1
representing the relative bias of Z •
e
An examination of these results clearly displays the similarity of
the behavior pattern of the relative bias with respect to the various
parameters--the same observations that were made in connection with the
relative bias of Z are repeated for the TE estimator.
e
is clear; for the conventional values of a, say a
=
However, one thing
.05, or even .10, the
bias of the TE estimator behaves very poorly in comparison with the bias
of Z for any of the
e
~
values being considered here.
But after normali-
zation as suggested earlier, the pattern of bias does not characterize
any fundamental difference in the two estimation procedures.
Our main
interest, however, is to study the effect on MSE that the two weighting
procedures exercise relative to each other, and we now address ourselves
to this question.
e
Table 7.
~
.1
.4
1.0
2.0
4.0
e
¥.
1
Values of PS2 E(Sl-T) for selected values of S2' p, v and a
v = 2
.2
I
.4
I
.6
I
.8
.2
I
I
.4
0.9250
0.9125
0.8500
0.6575
0.2362
0.9167
0.9167
0.8667
0.7125
0.3267
0.9250
0.9188
0.8912
0.7988
0.5150
0.9000
0.8375
0.3800
0.0050
0.0000
0.8500
0.8375
0.7100
0.4050
0.0438
0.8500
0.8375
0.7300
0.4475
0.0631
0.8500
0.8417
0.7583
0.5225
0.1175
0.8625
0.8500
0.8000
0.6488
0.2806
0.7500
0.6125
0.1400
0.0000
0.0000
0.9250
0.8438
0.4375
0.0012
0.0000
.1
.4
1.0
2.0
4.0
0.2000
0.1500
0.0550
0.0000
0.0000
0.2000
0.1625
0.0625
0.0025
0.0000
0.1833
0.1667
0.0817
0.0067
0.0000
0.1875
0.1781
0.1188
0.0027
0.0000
0.2000
0.1125
0.0050
0.0000
0.0000
.6
0.9167
0.8667
0.5433
0.0408
0.0000
===
0.7500
0.6312
0.1800
0.0000
0.0000
u
v = 18
I
I
I
.2
0.9250
0.8906
0.7088
0.2094
0.0000
0.9000
0.7000
0.0500
0.0000
0.0000
0.9000
0.7312
0.0800
0.0000
0.0000
0.9167
0.7792
0.1667
0.0000
0.0000
0.9250
0.8469
0.4150
0.0006
0.0000
0.7625
0.7094
0.4375
0.0519
0.0000
0.7000
0.4125
0.0050
0.0000
0.0000
0.7250
0.4438
0.0150
0.0000
0.0000
0.7333
0.5083
0.0417
0.0000
0.0000
0.7375
0.6031
0.1650
0.0001
0.0000
0.1875
0.1594
0.0488
0.0000
0.0000
0.2000
0.0625
0.0000
0.0000
0.0000
0.2000
0.0688
0.0000
0.0000
0.0000
0.1833
0.0917
0.0017
0.0000
0.0000
0.1875
0.1281
0.0012
0.0000
0.0000
.4
.6
I
.8
.8
= .05
a
0.9000
0.9125
0.8400
0.6250
0.1938
a
v = 8
a
.1
.4
1.0
2.0
4.0
e
a
0.2000
0.1188
0.0075
0.0000
0.0000
.10
0.7667
0.6625
0.2733
0.0083
0.0000
=1
0.1833
0.1333
0.0167
0.0000
0.0000
a Some minor discrepancies from the monotonic behavior with respect to A appear in these values,
due to a combination of the numerical approximation and rounding errors. Toro (1968, Table 14, p. 60)
presented absolute values of the bias instead of relative bias
VI
co
59
Comparison of Relative MSE
22
The values of MSE(Z ) for the chosen combinations of parameters v, p
e
and 6 were obtained using the results in (60) through (63).
2
The same
subroutine for numerical integration was employed to evaluate the necessary integrals.
Tables 9, 10 and 11 contain the values of MSE(Z )
e
relative to MSE(b ) for the five different choices of ; mentioned
l
earlier.
From formula (60), it is clear that the relative MSE depends
explicitly on p2 also, apart from A.
curve for each value of p.
Hence, there will be a separate
From the tables and Figures 3, 4 and 5, we
can observe the following:
(i) For a fixed p, v and ;, the relative MSE first increases with A
(or 6 ), reaches a maximum and then steadily decreases to converge
2
asymptotically to 1 as A approaches infinity.
1
to the right of A = 2'
A
= 21
The maximum always lies
However, the relative MSE is smaller than 1 at
for all choices of ; being considered here.
Hence, because of
continuity of the MSE function, there exists a neighborhood, its length
depending upon the particular choice of ;, in which.MSE(Z) is smaller
e
than both MSE(b ) and MSE(6 ).
1
1
(ii) The relative MSE is highly sensitive to the.particular choice
of;.
At the point A
= 0,
it is a monotonic increasing function of ;,
so that the relative MSE curve starts off closer to 1, the larger the
value of ;, as Figures 3, 4 and 5 clearly indicate.
In fact, throughout
the range of A, theMSE curve is flatter (thus remaining closer to the
22Table 8 below gives minimum MSE (~.~., MSE of the parametric
estimator
compared.
*
Z~)
against which the values of Tables 9 through 11 should be
e
Table 8.
~
~
e
~
•
*
Values of the ratio MSE(ZQ,)!MSE(b ) for selected values of 13 , p and
1
2
.20
I'
.40
I·
.50
I
.60
1.70
\!
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
0.9615
0.9752
0.9884
0.9917
0.9939
0.9959
0.9976
0.9984
0.8452
0.8959
0.9492
0.9633
0.9726
0.9813
0.9889
0.9927
0.7573
0.8311
0.9144
0.9375
0.9530
0.9677
0.9808
0.9873
0.9635
0.9842
0.9944
0.9962
0.9973
0.9982
0.9990
0.9993
0.8524
0.9317
0.9749
0.9830
0.9878
0.9920
0.9954
0.9970
0.7674
0.8864
0.9569
0.9706
0.9788
0.9860
0.9919
0.9948
0.6617
0.8221
0.9294
0.9513
0.9648
0.9766
0.9865
0.9912
\!
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
0.9664
0.9902
0.9970
0.9980
0.9986
0.9991
0.9995
0.9997
0.8630
0.9566
0.9864
0.9910
0.9936
0.9959
0.9977
0.9985
0.7826
0.9265
0.9764
0.9844
0.9889
0.9928
0.9959
0.9974
0.6809
0.8819
0.9608
0.9739
0.9815
0.9879
0.9931
0.9956
.80
\!
I
.90
I
.96
2
0.6490
0.7466
0.8636
0.8989
0.9232
0.9467
0.9680
0.9788
\!
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
=
I
e
0.4472
0.5605
0.7347
0.7955
0.8402
0.8861
0.9297
0.9529
0.3691
0.4798
0.6669
0.7377
0.7918
0.8491
0.9053
0.9360
0.1961
0.2778
0.4551
0.5398
0.6133
0.7011
0.7995
0.8591
0.0813
0.1225
0.2325
0.2984
0.3651
0.4597
0.5912
0.6886
0.4611
0.6691
0.8520
0.8953
0.9229
0.9481
0.9696
0.9802
0.3822
0.5939
0.8063
0.8609
0.8965
0.9297
0.9584
0.9728
0.2051
0.3788
0.6345
0.7207
0.7832
0.8464
0.9058
0.9371
0.0855
0.1811
0.3863
0.4834
0.5671
0.6666
0.7772
0.8438
0.4030
0.7026
0.8859
0.9220
0.9437
0.9628
0.9785
0.9861
0.2197
0.4962
0.7640
0.8313
0.8748
0.9152
0.9500
0.9673
0.0926
0.2632
0.5400
0.6411
0.7171
0.7965
0.8733
0.9147
=8
= 18
0.4828
0.7656
0.9148
0.9481
0.9586
0.9728
0.9844
0.9899
0\
o
~
e
Table 9.
:sJ
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
-,
"
Values of the ratio
.20
I
.40
MSE(Ze)/MSE~b1)
I
.50
I
for selected values of A and
I
.60
~
0.9781
0.9889
1.0104
1.0142
1.0173
1.0241
1. 0188
1.0112
0.9120
0.9496
1. 0309
1.0655
1.0665
1.0968
1. 0826
1.0544
0.8622
0.9150
1.0347
1.0893
1.0983
1.1489
1.1373
1.0978
0.8008
0.8666
1.0236
1.1006
1.1626
1.1570
1. 2102
1.1644
0.9846
0.9927
1. 0073
1.0119
1.0138
1.0124
1.0069
1.0028
0.9378
0.9664
1. 0232
1.0433
1.0534
1.0529
1. 0332
1.0155
0.9023
0.9429
1.0274
1.0604
1.0799
1.0851
1.0594
1.0305
0.9875
0.9943
1.0062
1. 0094
1. 0102
1.0084
1.0039
1. 0013
0.9497
0.9740
1.0199
1.0345
1.0404
1.0369
1. 0197
1. 0076
0.9211
0.9556
1.0243
1.0489
1.0619
1.0609
1.0367
1.0158
0.8859
0.9290
1.0213
1.0589
1.0829
1.0912
1.0645
1.0323
I
~,
v
.80
..
e
=2
I
.90
I
.96
= .25
0.6423
0.7104
0.8950 a
1.0050
1.1128
1.2518
1. 2790
1.3844
0.5444
0.5909
0.7271
0.8178
0.9166 a
1. 0687
1.2742
1.3348
0.4795
0.5016
0.5699
0.6187
0.6756
0.7730
0.9541 a
1.1374
0.7458
0.7988
0.9366 a
1.0142
1.0855
1.1666
1.2204
1.1945
0.6761
0.7124
0.8166
0.8839
0.9554 a
1.0600
1.1986
1.2690
0.6298
0.6471
0.7002
0.7376
0.7808
0.8535
0.9837 a
1.1071
0.7946
0.8398
0.9552
1.0181
1.0740
1.1340
1.1621
1.1303
0.7382
0.7692
0.8574
0.9136
0.9723 a
1.0562
1.1605
1.2041
0.7007
0.7155
0.7607
0.7925
0.8289
0.8897
0.9965 a
1.0942
= .50
0.7773
0.8332
0.9725 a
1.0711
1.1083
1.1711
1.1903
1.1468
0.8586
0.9094
1.0220
1.0711
1.1047
1.1240
1.0995
1.0597
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.75
0.6865
0.7584
0.9471 a
1.0537
1.1529
1.2069
1. 3373
1. 3212
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.
."
e
= .70
0.8201
0.8677
0.9836a
1.0422
1.0901
1.1342
1.1349
1.0933
0\
I-'
...
e
e
~
••
• 9
e
Table 9 (continued)
e:s;:}
.20
I
.40
I
.50
I
I
.60
~
.1
0.9903
0.9958
1.0050
1. 0069
1.0073
1.0054
1.0019
1.0004
.4
.8
1.0
1.2
1.5
2.0
2.5
0.9614
0.9812
1.0165
1.0262
1. 0292
1.0242
1.0108
1.0034
0.9393
0.9675
1.0206
1.0380
1.0452
1.0412
1.0210
1.0074
.1
0.9948
0.9981
1.0029
1.0036
1. 0033
1.0020
1. 0005
1.0001
.4
.8
1.0
1.2
1.5
2.0
2.5
a
0.9792
0.9910
1. 0101
1.0140
1.0139
1.0097
1.0029
1.0005
0.9674
0.9842
1.0134
1.0209
1.0224
1. 0173
1.0062
1.0014
0.9528
0.9739
1.0142
1.0271
1.0323
1.0286
1.0130
1.0037
.80
I
.90
I
.96
= 1.0
0.9122
0.9473
1.0197
1.0471
1.0622
1.0637
1.0389
1.0164
~
I
.75
=
0.8617
0.9005
0.9928 a
1.0374
1.0720
1.0995
1.0905
1.0552
0.8420
0.8789
0.9713
1.0200
1.0618
1.1030
1.1133
1.0819
0.7985
0.8239
0.8954
0.9403
0.9865 a
1.0505
1.1246
1.1478
0.7705
0.7817
0.8187
0.8445
0.8740
0.9226
1.0062
1. 0797
0.9254
0.9489
1.0020
1.0255
1.0418
1.0507
1.0374
1.0176
0.9148
0.9372
0.9909
1.0174
1.0384
1.0555
1.0509
1.0297
0.8912
0.9067
0.9494
0.9753
1.0011
1.0350
1.0683
1.0710
0.8755
0.8830
0.9054
0.9208
0.9382
0.9665
1.0131
1.0509
2.0
A
Denotes that MSE(Ze) is smaller than both MSE(b ) and MSE(Sl)
1
0\
N
e
...
.'
e
~
Table 10 •. Values of the ratio MSE(Ze)/MSE(bi) for selected values of A and
;sj
.20
I
.40
I
.50
I
I
.60
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
0.9750
1.0010
1.0269
1.0259
1. 0192
1.0097
1. 0019
1. 0005
0.8983
0.9933 a
1.1050
1.1084
1.0891
1.0503
1.0126
1.0021
0.8400
0.9765 a
1.1558
1.1730
1.1520
1.0944
1.0270
1.0055
0.7676
0.9405 a
1.2017
1.2478
1. 2375
1.1684
1.0597
1.0147
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
0.9827
1.0019
1.0154
1.0105
1.0067
1.0020
1.0000+
1.0000+
0.9302
1.0016
1.0605
1.0512
1.0328
1.0123
1.0016
1.0001
0.8896
0.9926 a
1.0938
1. 0863
1.0611
1.0260
1.0040
1.0005
0.8393
0.9718 a
1.1280
1.1331
1.1056
1.0541
1.0102
1.0011
~
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
0.9865
1.0025
1.0105
1.0070
1.0035
1.0008
1.0000+
1.0000+
0.9451
1.0040
1.0439
1.0331
1.0188
1.0055
1.0004
1.0000+
0.9133
0.9987 a
1.0694
1.0577
1.0363
1.0126
1.0011
1. 0001
0.8736
0.9835 a
1.0978
1.0926
1.0666
1.0284
1.0035
1.0003
.75
I
~,
.80
v
I
e
=8
.90
I
.96
= .25
0.6304
0.8260 a
1.2049
1.3309
1.3864
1.3602
1.2008
1.0794
0.5767
0.7642
1.1612
1. 3197
1.4143
1. 4384
1. 2988
1.1433
0.4560
0.5879
0.9278 a
1.1153
1.2842
1.4689
1.5602
1.4583
0.3748
0.4388
0.6263
0.7508 a
0.8862 a
1.0932
1. 3935
1.5808
0.7055
0.8527
1.1278
1.2120
1.2423
1.2101
1.0961
1.0284
0.6206
0.7256
0.9798 a
1.1065
1.2080
1. 2938
1.2774
1.1725
0.5633
0.6148
0.7618
0.8560
0.9547 a
1.0966
1. 2738
1. 3567
0.7680
0.8920
1.1089
1.1655
1.1766
1.1375
1.0505
1.0112
0.7007
0.7899
0.9985 a
1.0969
1.1705
1.2222
1.1849
1.0988
0.6551
0.6991
0.8229
0.9007
0.9805 a
1.0916
1.2188
1.2556
= .50
0.7433
0.8958
1.1485
1.2057
1.2089
1.1541
1.0536
1.0106
= .70
0.7979
0.9260
1.1212
1.1549
1.1459
1.0948
1. 0252
1.0040
0'\
w
e
~.
,
..
e
•
•
e
Table 10 (continued)
:s:l
.20
I
.40
I
.50
I
I
.60
~
.1
.4
0.9904
1.0019
1.0067
1.0038
1.0019
1. 0003
1.0000+
1.0000+
.8
1.0
1.2
1.5
2.0
2.5
0.9597
1.0050
1.0294
1.0202
1.0101
1.0025
1.0001
1.0000+
0.9360
1.0028
1.0485
1.0358
1.0200
1.0054
1.0003
1.0000+
0.9065
0.9930 a
1.0704
1.0602
1.0391
1.0134
1.0013
1.0001
~
0.9950
1. 0016
1.0027
1.0012
1.0004
1.0000+
1.0000
1.0000
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
a
0.9797
1.0043
1. 0123
1.0067
1.0025
1. 0003
1.0000+
1.0000
0.9678
1.0039
1.0208
1.0129
1.0057
1.0010
1.0000+
1.0000
0.9529
1.0000+
1.0319
1.0233
1.0123
1.0030
1. 0001
1.0000+
=
I
.75
.80
I
.90
I
.96
1.0
0.8501
0.9516
1.0929
1.1095
1.0955
1.0540
1.0160
1.0011
0.8276
0.9266
1. 0871
1.1213
1.1206
1. 0835
1.0241
1.0040
0.7775
0.8483
1.0104
1.0822
1.1316
1.1580
1.1147
1.0521
0.7432
0.7788
0.8771
0.9376
0.9986a
1.0802
1.1646
1.1768
0.9128
0.9860
1.0468
1.0574
1.0507
1.0287
1.0053
1.0005
0.8871
0.9275
1.0137
1.0472
1.0671
1.0709
1.0408
1.0137
0.8695
0.8897
0.9446
0.9772
1.0089
1. 0478
1.0809
1.0758
= 2.0
0.9242
0.9806
1.0472
1.0489
1.0371
1.0164
1.0018
1.0001
A
Denotes that MSE(Ze) is smaller than both MSE(b ) and MSE(S1)
1
0~
e
Table 11.
:sJ
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
..
,
e
••
Values of the ratio MSE(Ze) /MSE(bi) fOT selected values of Aand
.20
I
.40
I
.50
I
I
.60
i;
0.9770
1. 0173
1. 0211
1.0115
1.0038
1.0000+
1.0000+
1.0000+
0.9025
1.0555
1.0975
1.0588
1.0269
1. 0050
1.0000+
1.0000+
0.8460
1.0694
1.1639
1.1084
1.0559
1. 0144
1.0007
1.0000+
0.9846
1.0115
1.0077
1.0019
1.0006
1.0000+
1.0000+
1.0000+
0.9328
1.0420
1.0387
1.0151
1.0050
1.0000+
1.0000+
1.0000+
0.8938
1.0555
1.0696
1.0316
1.0103
1.0012
1.0000+
1.0000+
0.8438
1.0589
1.1178
1.0640
1.0256
1.0038
1.0000+
1.0000+
i;
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
0.9876
1.0097
1.0043
1. 0011
1.0002
1.0000+
1.0000+
1.0000+
0.9491
1.0346
1.0224
1.0070
1. 0015
1.0001
1.0000+
1.0000+
0.9186
1.0474
1.0425
1.0156
1.0040
1.0003
1.0000+
1.0000+
0.8795
1.0533
1.0757
1.0348
1.0111
1.0012
1.0000+
1.0000+
I
.80
\!
I
= 18
.90
I
.96
= .25
0.6317
0.9843 a
1. 3859
1. 3851
1.3011
1.1532
1.0281
1.0036
0.7721
1.0666
1.2510
1.1895
1.1114
1.0358
1.0026
1.0000+
i;
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.75
i;,
e
0.5745
0.9201 a
1. 4003
1. 4557
1.4032
1. 2484
1. 0648
1.0108
0.4450
0.6992
1. 2327
1. 4371
1.5511
1. 5587
1. 3456
1.1426
0.3559
0.4820
0.8828 a
1.0299
1.2240
1.4621
1.6498
1.5998
0.7041
0.9712 a
1.2448
1. 2275
1.1613
1.0662
1.0072
1.0000+
0.6115
0.8151
1.1835
1.2872
1.3145
1. 2547
1.0992
1.0230
0.5476
0.6518
0.9169 a
1.0597
1.1844
1.3119
1.3492
1.2505
0.7710
a
0.9919
1.1822
1.1529
1.0959
1.0314
1.0022
1.0001
0.6984
0.8694
1.1562
1.2220
1.2255
1.1618
1.0487
1.0082
0.6480
0.7365
0.9551 a
1.0674
1.1604
1. 2447
1.2423
1.1517
= .50
0.7418
1.0131
1.2180
1.1731
1.1035
1.0325
1.0021
1.0000+
= .70
0.8024
1.0231
1.1555
1.1091
1.0565
1.0134
1.0004
1.0000+
0'\
\J1
e
-
eo
•
•
e
Table 11 (continued)
G
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
.20
I
.40
I
.50
I
I
.60
I
.75
.80
I
.90
I
.96
t,; = 1.0
0.9904
0.10077
1.0019
1.0000+
1.0000+
1.0000+
1.0000+
1.0000+
0.9647
1.0269
1.0118
1.0034
1.0000+
1.0000+
1.0000+
1.0000+
0.9420
1.0379
1.0229
1.0064
1.0019
1.0000+
1.0000+
1.0000+
0.9142
1.0435
1.0448
1.0166
1.0038
1.0000+
1.0000+
1.0000+
0.8583
1.0272
1.1025
1.0622
1.0272
1.0049
1.0001
1.0000+
0.8359
1.0058
1.1253
1.0936
1.0511
1.0130
1.0007
1.0000+
0.7824
0.9166
1.1229
1.1590
1.1491
1.0937
1.0215
1.0025
0.7456
0.8157
0.9836 a
1.0653
1.1288
1.1778
1.1545
1.0834
0.9081
1.0047
1.0545
1.0343
1.0152
1.0026
1.0045
1.0000
0.8781
0.9563
1.0623
1.0730
1.0612
1.0313
1.0602
1.0003
0.8570
0.8987
0.9936 a
1.0363
1.0662
1.0832
0
1.0255
t,; = 2.0
.1
.4
.8
1.0
1.2
1.5
2.0
2.5
0.9952
1.0035
1.0005
1.0001
1.0000+
1.0000
1.0000
1.0000
a
0.9800
1.0132
1. 0034
1.0005
1.0000+
1.0000
1.0000
1.0000
0.9679
1. 0190
1.0073
1.0015
1.0002
1.0000+
1.0000
1.0000
0.9523
1.0231
1.0152
1.0042
1.0007
1.0000+
1.0000+
1.0000
0.9209
1.0156
1.0416
1.0206
1.0070
1.0007
1.0000+
1.0000
A
Denotes that MSE(Ze) is smaller than both MSE(b ) and MSE(Sl)
1
0'
0'
e
e
...
•
e
•
R MSE
~
/~A
.
Sl
1.30
----
--------- ---
/
1.20
••• 8
1.10
-
••• 8
~
~
~
E,;
=
=
=
=
=
025
.50
.70
LO
2.0
--- ---
_ _ -:- :-:-=: -: -: "7...- ~~.: ~ ~ -.--:-.-.-.__~ ~-~-.~ -.~ ~.-.-.-.:; ~~ ~ ::: ~.-.-.=. ~ -:- ~~:-:- .-.-=; ~ --: ::._:-.:;
:---
1.00
rl
b
1.90
1
.80
.70
.60
.50
.40
0
0
.4 .5
Figure 3.
.8
1.2
1.6
Relative MSE (rl
2.0
ze /rl b 1 )
2.4
2.8
curves for different
3.2
E,;
3.6
4.0
values and p = .40,
4.4
v =
4.8
8
0\
-...J
e
e
....
•
e
4
1.40
QA
f\
1.30
1.20
---
--
--- --- --- -
--.
-- --
.:= ~--=:=------- - ::: ~---=-:::..::..:::.::::. : . .---..::..:::.:: =-=---- - -::::::..
1.10
-
1.00
..
Q
b
.90
.80
.70
........
. 60
.50
.40
0
t;
t;
t;
t;
t;
1
= .25
=
=
=
=
.50
.70
1. 00
2.00
I
1
'-
0
.4 .5
Figure 4.
.8
1.2
1.6
2.0
Relative MSE (Qz /Q
e
b
2.4
2.8
) curves for different
3.2
t;
3.6
values and p
4."
= .60,
4.4
v
4.8
=8
1
0\
00
e
.
e
-.
e
\
1.60
1.50
nA
81
I
1.40
--
I
1.30
1.20
/
----------------- -- ---. ------- -- -- - --- ------- ---_._---------------
",-
----
---------
--
--- -...---. .
1.10
~~......././
/'
-
.
/
. ...
'"
......
1.00
.90
~1
I"
.80
----
~
~
=
=
~.=
~
=
.70
~
=
.25
.50
.70
1.00
2.00
.60
.50
.40
0
L
0
:4
Figure 5.
.8
1.2
Relative MSE
1.6
2.0
<nz /nb
e
2.4
2.8
3.2
3.6
4.0
4.4
4.8
) curves for different ~ values and p = .90, v = 8
1
0\0
70
horizontal line in the figures), the larger the vallile
of~.
Implicit in
this result is the fact that there is one and only one intersection point
between curves with any two different values
p and v, choice of a smaller
~
Also, for each fixed
of~.
appears to result in a larger A interval
over which Ze has advantage over both b
1
and Sl' but this gain is attained
at the price of having a larger MSE from the point of intersection onwards,
and a slower rate of asymptotic convergence to Q •
b
1
(iii) The influence of p2, the index of multicollinearity, on the
relative MSE, also bears considerable interest.
At first, it seems that
this influence can be studied by fixing the value of S2 over the range of
However, since A is a monotone decreasing function of p2 (for fixed S2)
p.
and since the relative MSE is (explicitly) a function of both A and p2,
such a procedure will only give a confounded effect on both A and p2.
Thus, for a fixed S2 (and other parameters), the RMSE is a monotonic
decreasing function of p2 for small values of A, is a convex function of
p2 in some intermediate range of A, and is monotonic increasing in p2
for large values of A.
The precise effect of p2 can be measured by comparing the values
of RMSE for approximately the same value of A (and, of course, all other
parameters) •
p
=
Thus, for example, for v = 8, A is .6000 for S2
.50, and A is •6080 for S
2
=
.8 and p2
=
.90 •
= .4
and
Thus, comparing for
each E; value, the values of RMSE for these combinations of S2 and p,
.
we are led to the following interesting observation:
The RMSE is smaller
~
for a larger p whenever both are smaller than 1; the reverse is true
whenever both are greater than 1.
This result follows from the fact
that, for a fixed A, the RMSE is a monotonic decreasing function of p2
71
whenever H(A) in (72) is negative, and monotonic increasing in p2 whenever
H(A) is positive.
Thus a higher multicollinearity content in the data
accentuates both the advantage and the disadvantage of using the CWF
procedure.
This is true for any choice of
~
value, although the differ-
ence becomes less and less significant, the higher the value
Note
of~.
that this observation again emphasizes the desirability of having some
prior information on A.
Table 12 below gives the values of RMSEs for
selective adjacent values of A corresponding to different values of p2.
The values of MSE(T) were obtained by Toro (1968) for selected
combinations of the parameters and for the three significance levels
described earlier.
relative to MSE(b ).
l
In Table 13 are presented the values of MSE(T)
Examination of these results verifies all the
conclusions obtained analytically in Chapter II, in addition to some
observations which could not be derived analytically.
Comparison of
these results with the ones for MSE(Z ) lead to the following important
e
observations:
(i) The general pattern of variation of the RMSE over the range of
A, ceteris paribus, is similar for both the TE and exponential weighting
procedures--it increases monotonically at first, reaches a maximum,
and then steadily decreases to approach unity asymptotically as A increases
without bounds.
However, the degree of control over the MSE in the two
procedures is dramatically reflected in the fact that while the MSE(T)
.
~
becomes larger than both MSE(b ) and MSE(Sl) in some neighborhood of
l
1
A=2
'
1
MSE(Ze) is smaller than both in some neighborhood of A = 2
irrespective of the particular choice of a and
~.
'
..
e
Table 12.
e
Relative MSEs of Ze' Sl and
*
Z~
••
•
•
e
for selected values of A over the range of values of p:
The co11inearity effect
A
MSE(Ze)/MSE(b,)
62
\)
p
A
F;
= .25
I
F;
=
.5
I
F;
=
.7
I
F;
= 1.0
MSE(Sl)
MSE-(b )
1
MSE(Z,Q,*
MSE(b )
1
.4
1.0
1.5
.4
1.0
.4
2
2
2
8
8
18
.20
.90
.96
.75
.96
.90
.3072
.3800
.3528
.3500
.3920
.3040
.9889
.8178
.7730
.8260
.7508
.6992
.9927
.8839
.8535
.8958
.8560
.8151
.9943
.9136
.8897
.9260
.9007
.8694
.9958
.9403
.9226
.9516
.9376
.9166
.9846
.8056
.7281
.8313
.8010
.6825
.9752
.5398
.4597
.6691
.4834
.4962
.8
1.2
.4
.8
2
2
8
18
,80
.90
.60
.96
.4608
.5472
.5120
.5018
.8950
.9166
.9405
.8828-
.9366
.9554
.9718
.9169
.9552
.9723
.9835
.9551
.9713
.9865
.9930
.9836
.9498
1.0765
1.0086
1.0033
.6669
.6133
.8221
.5400
.8
2.0
.4
.8
1.2
.4
2
2
8
8
8
18
.75
.96
.50
.90
.96
.80
.5600
.6272
.6000
.6080
.5645
.5760
.9471
.9541
.9765
.9278
.8862
.9201
.9725
.9837
.9926
.9798
.9547
.9712
.9836
.9965
.9987
.9985
.9805
.9919
.9928
1.0062
1.0028
1.0104
.9986
1.0058
1.0675
1. 2344
1.0500
1.1750
1.1189
1. 0973
.7347
.5912
.8864
.6345
.5671
.7026
1.0
.4
.4
.8
1.0
1.5
.4
1.5
1.0
2
8
18
2
2
2
8
8
18
.80
.40
.75
.60
.75
.90
.20
.96
.96
.7200
.6720
.7000
.8192
.8750
.8550
.7680
.8820
.7840
1.0050
.9933
.9843
1. 0236
1.0537
1. 0687
1.0010
1.0932
1.0299
1.0142
1.0016
1. 0131
1.0220
1. 0457
1.0600
1. 0019
1.0966
1.0597
1.0181
1.0040
1.0231
1.0213
1.0422
1. 0562
1.0025
1.0916
1.0674
1.0200
1.0050
1.0272
1.0206
1.0374
1.0505
1. 0019
1.0802
1.0653
1.2816
1.0550
1.2250
1.2298
1.4219
1. 5751
1.0214
1.7041
1.5235
.7377
.9317
.7656
.8636
.7955
.7011
.9842
.6666
.6411
......
'"
e
'.
Table 13.
~
.1
.4
1.0
2.0
4.0
-
~
v
.20
I
.40
=2
I
v
.60
I
.80
.20
I
I
=
a
0.8568
0.9375
1.3441
2.3790
3.0948
0.6759
0.8116
1.5258
3.5049
5.9447
0.4176
0.5544
1.2932 0
3.5829
8.6521
0.9693 0
1.0269
1.1612
1.0077
1.0000+
0.9677
0.9908
1. 0906
1. 2596
1.1137
0.8669
0.9442
1.3138
1. 9825
1.5860
0.6989
0.8320
1.4926
2.9237
2,8623
0.4594
0.5948
1. 2975 0
3.2085
5.3615
0.9693
1.0269 0
1. 0749
1.0000+
1.0000+
0.8655
1.0756 0
1.6471
1.0756
1.0000+
e
\
0.9711
0.9980
1.0450
1.0081
1. 0013
0.9344
0.9805
1.0958
1. 0369
1.0020
0.8799
0.9303
1.1060
1.1550
1. 0023
0.9981
1. 0077
1. 0000+
1.0000+
1.0000+
v
I
.60
I
= 18
I
I
.80
.20
0.4356
0.7991
2.4622
3.3081
1.0079
0.9597
1.0749
1.0557
1.0000+
1.0000+
0.8739
1.2605
1.2773
1.0000+
1.0000+
0.7170
1. 4085 0
1. 9846
1.0000+
1.0000+
0.4608
1. 1879 0
3.2181
1.1591
1.0007
0.5328
0.7490
1.0819 0 0.8855
1. 7414
2.0734
1.6523
1.0499
1.0000+ 1.0007
0.9789
1.0557
1.0000
1.0000+
1.0000+
0.9076
1.1933
1.0588
1.0000+
1.0000+
0.7811
1.3444
1.2676
1.0000+
1.0000+
0.5760
1. 2239 0
2.0086
1.0079
1.0007
0.9981
1.0173
1.0000+
1.0000+
1.0000+
0.9748
1.0420
1.0084
1.0000+
1.0000+
0.9475
1.0883
1.0115
1.0000+
1.0000+
0.8927
1. 0871
1. 0871
1.0007
1.0007
.40
.60
.80
.05
0.6914
1.0499 0
2.3111
1.4789
1.0000+
. .10
=
0.8908
1.0765 0
1. 3190
1.0084
1.0000+
u
0.9908
0.9985
1.0100
1.0023
1. 0000+
=8
.40
a
.1
.4
1.0
2.0
4.0
,
Values of the ratio MSE(T)/MSE(b ) for selected values of A, v and aa
1
0.9639
0.9869
1.1022
1. 3786
1. 4938
.1
.4
1.0
2.0
4.0
.'
a
=1
0.9748
1.0252
1. 0168
1.0000+
1.0000+
0.9411
1.0371 0
1.0627
1.0000+
1.0000+
0.8819
1.0007 0
1.1699
1.0115
1.0007
a Toro (1968, Table 15) presented absolute values of MSE(T) instead of the ratio
o
A
Denotes that MSE(T) is 1arger- than both MSE(b ) and MSE(Sl)
1
'"w
74
(ii) For any particular choice of a and
~,
the difference between
MSE(T) and MSE(Z ) (i.e., either the supremacy of T over Z in the MSE
e
-e
sense or vice versa), becomes sharper as p2 increases (for a fixed v),
and becomes milder as v increases (for a fixed p).
values of v and p,
as A gets very large
approach MSE(b ) •
l
.
Also, whatever the
this difference becomes less and less significant
(~.~.,
as both MSE(Z ) and MSE(T) asymptotically
e
75
CHAPTER V
SUMMARY AND CONCLUSIONS
Summary
This is a problem dealing with the implications of using a preliminary
test on subsequent estimation of a regression paramerer(s).
By a prelim-
inary test is meant here a test on a nuisance parameter which precedes
estimation of the parameter of interest.
Two different procedures for
the use of a test are compared for efficiency in the final estimation.
One procedure involves dependence of the ultimate estimator on the outcome of the preliminary test, the resulting estimator being called the
TE estimator.
The second procedure is characterized by one-to-one
functional dependence on the test-statistic itself, resulting in the
continuous weight function (CWF) estimator.
All the results are illus-
trated with the linear regression model containing two nonstochastic
regressors.
Given the objective of obtaining a "good" estimator of (31'
two somewhat interrelated situations, viz., that arising from specification error, and one from multicollinearity, are envisaged.
Both
these situations, however, can be handled with the same testing procedure,
with only a change in the significance level (a.), one corresponding to
the usual t-test, and the other to the MSE test.
The analytical objective of this investigation is to examine, first
..
individually, the influence of the various, parameters involved on the
bias and MSE of both the TE estimator and the CWF estimator; subsequently,
:
via a comparative analysis of some numerical results, to study the
relative degree to which each estimation procedure can control the MSE
over the range of the nuisance parameter.
76
In Chapter II, the analyticaL theory for.the TE procedure has been
presented.
The density function and the moment generating function of
the TE estimator of Sl has been derived, whence its bias and MSE were
obtained.
The role of the various parameters involved, viz., A, p and
a, in determining the behavior of bias and MSE was studied analytically
and important generalizations pertaining to the same were obtained, as
far as theoretically tractable.
In Chapter III, the continuous weighting procedure was introduced as
an alternative to the TEprocedure, and the relevant theory was developed
in broad outlines.
The various alternatives available for choosing a
specific weight function within the general framework were discussed,
and ultimately the exponential family of weight functions was chosen
for detailed investigation largely because of its theoretical tractability.
Conclusions on bias and MSE of the resulting estimator of Sl
were obtained from theoretical considerations .. to the extent such
analysis was possible.
In Chapter IV, some numerical results were obtained for selected
combinations of the various parameters involved both for verification
of the generalization derived analytically and to throw further light
on the relative merits of the two procedures.
Conclusions
From a combination of analytical results and numerical observations,
:
the following conclusions of major significance emerged:
(i) The bias in both the TE estimator and the CWF estimator displays
similar pattern of behavior over the range of the (unknown) nuisance
parameter (denoted by A); it increases first, reaches a maximum, and.
77
and then steadily decreases to approach zero as A approaches infinity.
However, in any comparable situation of the two estimation procedures
(i.~.,
after normalization in some sense), both the rate of increase and
the rate of asymptotic convergence to zero are different for the TE
estimator and the CWF estimator.
(ii) The MSE of the TE estimator lies between the MSEs for the two
simple estimators, viz., b
l
and Sl' over the two extreme ranges of the
nuisance parameter A, say, [O,A
O
]
and [Al,oo].
However, for every value
of a not equal to zero or one, there exists an intermediate range of A
over which the MSE of the TE estimator is larger than both MSE(b ) and
l
MSE(Sl).
Thus, over this range of A, the TE procedure is less desirable
in the MSE sense
tha~
even an arbitrary choice of b
l
or Sl as the
estimator of Sl.
(iii) On the other hand, for any typical member of the CWF estimator
of Sl' there exists an intermediate range of A over which its MSE is
smaller than both MSE(b ) and MSE(Sl).
l
Thus, if a large prior weight
can be attached to intermediate ranges of A (as against the two extreme
ranges), then the choice will obviously be in favor of CWF estimation
procedure.
(iv) Further, numerical results showed that on the whole, the CWF
procedure exercises a distinctly greater degree of control on the MSE,
compared with the TE procedure, over the entire range of A.
Suggestions for Further Research
The main emphasis of this thesis has clearly been to demonstrate
that the conventional approach to the selection of variables in the
framework of incompletely specified (linear regression) models, viz.,
78
letting the choice of the correct model be made by the outcome of
significance tests of "nuisance" parameters--is, at best, woefully inadequate in terms of efficient use (in the MSE sense) of the available
information, and can be perceptively improved upon by adopting the
alternative approach represented by the CWF procedure.
The particular
alternative proposed in this study, however, constitutes only one of
several possible ways of maximizing the use of potential information
contained in the data.
It is, for example, a natural and logical ques-
tion to ask whether the weight function in the CWF procedure must
necessarily be taken as a function of the test-statistic itself, or can
more usefully be taken as some other function of the sample
observations.
The particular form of the weight function chosen in this
investigation is determined more by considerations of mathematical
convenience and tractability than by any specific optimality criteria.
Since the results in this study show that considerable room for improvement in the estimation still exists, it would be worthwhile to explore
the feasibility of adopting simpler and more effective weight functions.
Finally, the statistical model considered in this study has been
considerably simplified, and is used more as a vehicle for illustrating
underlying ideas and concepts than as a representation of realistic
situations, both in terms of scope and dimensions.of the problem.
Consequently, extensions of the results to the case of multidimensional
problems and a synthesis of the analytical structure to include both
prediction and estimation objectives for general linear models would
definitely contribute to consolidation and crystallization of the ideas.
79
LIST OF REFERENCES
Bancroft, T. A. 1944. On biases in estimation due to the use of
preliminary tests of significance. Ann. Math. Stat. 15:190-204.
Bancroft, T. A. 1964. Analysis and inference for incompletely
specified models involving the use of preliminary tests of
significance. Biometrics 20:427-442.
Bennett, B. M. 1952. Estimation of means on the basis of preliminary
tests of significance. Ann. Inst. Stat. Math. 4:31-43.
Graybill, F. A. 1961. An Introduction to Linear Statistical Models,
Vol. I. McGraw-Hill Book Company, Inc., New York.
Gun, A. M. 1965. The use of a preliminary test for interactions in
the estimation of factorial means. Institute of Statistics,
Mimeo Series No. 436, North Carolina State University, Raleigh, N. C.
Huntsberger, D. V. 1955. A generalization of a preliminary test
procedure for pooling data. Ann. Math. Stat. 26:734-743.
Kitagawa, T. 1950. Successive processes of statistical inferences.
Mem. Fac. Sc., Kyushu University, Series A 15:139-180.
Kitagawa, T. 1963. Estimation after preliminary tests of significance.
University of California Publications in Statistics 3(4):147-186.
Larson, H. J. and T. A. Bancroft. 1963. Sequential model building
for prediction in regression analysis. J. Ann. Math. Stat. 34:
462-479.
Mosteller, F.
242.
1948.
On pooling data.
J. Amer. Stat. Assoc. 43:231-
Rao, C. R. 1965. Linear Statistical Inference and Its Applications.
John Wiley and Sons, Inc., New York.
Sawa, T. 1968. Selection of variables in regression analysis.
Riron-Keizai Gaku 19:53-63.
Toro-Vizcarrondo, C. E. 1968. Multicollinearity and the mean square
error criterion in multiple regression: A test and some sequential
estimator comparisons. Unpublished Ph.D. thesis, Department of
Experimental Statistics, North Carolina State University at
Raleigh. University Microfilms, Ann Arbor, Michigan.
80
Toro-Vizcarrondo, C. E. and T. D. Wallace. 1968. A test of the mean
square error criterion for restrictions in linear regression.
J. Amer. Stat. Assoc. 63:558-572.
Wallace, T. D. 1964. Efficiencies for stepwise regressions.
Stat. Assoc. 59:1179-1182.
J. Amer.
81
APPENDIX
FORTRAN PROGRAMS FOR THE EVALUATION OF
NUMERICAL INTEGRALS INVOLVED IN THE BIAS AND MSE OF Z
e
Fortran Listing
2
REAL FUNCTION G*8 (U)
IMPLICIT REAL*8(A-H,O-Z)
COMMCN CEE
REAL *8 1,NO
X=U*C+U*C/(I.DO-U)
IF(X.LT.150)GO TO 1
WRITE(3,2) X,A,B,C,D,U
FORMAT(' X = ',GI5.6,'A=',GI5.6,'B=' ,GI5.6 ,'C=' ,GI5.6,'D=',GI5.6
l'U=' ,GI5.6)
c=o.
1
D=O.
X=l.+c+D
G=(U/(I.DO-U»**A*(I.DO-U)**B(DEXP(X)
RETURN
ENTRY INIT(NU,R,B2,L)
INIT=1
RETURN
ENTRY Il(J)
A=(NU+l)/2
B=-.5
C=L
D=-NU*DFE
Il=1
RETURN
ENTRY 12 (J)
A= (NU+l) /2
B=-.5
C=L
D=-NU*DEE*2
12=1
RETURN
ENTRY 13 (J)
A=(NU+l)/2
B=.5
C=L
D=-NU*DEE
13=1
RETURN
ENTRY
A=(NU+3)/2
B=.5
C=L
D=-NU*DEE*2
14=1
RETURN
END
82
Fortran Listing
•
10
o
1
11
REAL FUNCTION F*8(K)
IMPLICIT REAL*8(A-H,0-Z)
DIMENSION X(37000)
EXTERNAL G
IF(K,EQ.l)ISW=1
1=0
EPS=.OOOOOIDO
CALL DQATR(.ODO,.999500
EPS
LOW
HIGH
EPS
IF (IER.EQ.O) RETURN
1=1+1
WRITE(3,I)IER,K,I
FORMAT(' ERROR -ACCURACY NOT REACHED','
IF(ISW.EC.C)RETURN
EPS=SPS*3
IF(I.LT.7)00TO 10
WRITE(3,1l)
FORMAT(' ROUTINE FAILLED')
ISW=O
RETURN
END
,37000, G ,F,IER
,X)
DIMENSION FUN
ERROR
IER=' ,15,'K=' ,15,'1=' ,IS
83
Fortran Listing
IMPLICIT REAL*8(A-H,O-7)
COMMON CEE
REAL*8 NU,L,K1,K2
REAL*8 NOFE(10)/ .5, .7, 2./
REAL*8 VB2(10)/
.1, .4, .8, 1., 1.2, 1.5, 2., 2.5/
REAL*8 VR (10)/.04, .16, .36, .64/
REAL*8 VNU(10)/2., 8., 18./
NDEE=3
NB2=8
NR=4
NNU=3
DO 999 IDEE=l,NDEE
DO 999 IB2=1,NB2
DO 999 IR =l,NR
DO 999 INU=l,NNU
DEE=VDEE (IDEE)
B2=VB2(IB2)
R =VR(IR)
NU=VNU(INU)
WRITE(3,1)B2,R,NU,DEE
FORMAT(' BETA=' ,FlO.5,' R=' ,FlO.5,' NU=' ,FlO.5, 'D=' ,FlO.5)
1
L=B2**2* (NU+2) *(1-R)/2
AU=NU/2
CO= (NU*CEE) **AU/DGAMMA(AU)
K1=DEXP(-L)*CO
K2=K1*2**AU
CALL INIT(NU,R,B2,L)
X=ll(l)
X1=F(1)X=I2(1)
X2=F(2)
X=13(1)
X3=F(3)
X=14 (1)
X4=F(4)
F1=K1*X1
F2=K2*X2
F3=Kl*X3
F4=K2*x4
WRITE(3,6)F1,F2,F3,F4
6
FORMAT(' INTEGRAL1=' ,G17.5,' INTEGRAL 2= ',Gl7.5,'INTEGRAL3='
1G27.5,' INTEGRAL4=' ,G27.5)
FUN1=1+R*(F2+2*L*F4+4*L*F3-2*F1)
WRITE(3,8)FUN1
8
FORMAT(' FUN1= ',G15.7)
999
CONTINUE
STOP
END