•
JACKKNIFING MAXIMUM LIKELIHOOD ESTIMATES
IN THE CASE OF A SINGLE PARAMETE~
By
R. A.
Ferg~son
and J. G. Fryer
D~partment of Biostatistics
University Qf North-Carolina at Chapel Rill
and
University of Exeter, England
Institute of Statistics Mimeo Series No. 891
SEPTEMBER 1973
The main purpose of this paper is to illustrate and quantify the
effect that jackknifing can have on the lower sampling moments of a
maximum likelihood estimate.
i.s also considered.
Estimation of variance of the two estimators
In addition,
,~e
check on things like
c~e
rate of
convergence of the disCtibutions of the two basic estimatoR to normality,
and the precision of approximate confidence intervals based on them.
,~c
First
correct existing series formulae for the lower order moments of the two
estimate:l and estimates of variance and give sorne extensions.
Gee how things work out in some examples.
Simulations
ar~
Then we
used (when
necessary) to check on the adequacy of the series approxi.mAtions.
~n
two of the three examples that we discuss, jackknifing can be considered
to be an improvem ~l\t on the n,aximum 1i.kelihood estimate on the whole, though
the degree of improvement varies a lot depending on the measure concerned.
In.the third illustration jackknifing al~aYG improves bias and usually
V8t'ianCp..
t.h~
But there is little \lias in the maximum likelihood estimatr.l anyway and
p" ..eentage changes in variance are usually small.
Tukcy's estimate of variance turns out to be
i.nf(~dor
In all thre.e examples
to simple alternatives.
- 1 -
1.
Introduc~i~
Despite the recent increase in the number of publications on the jackknife
relatively little
~\lantit~~_~~~ even
now seems to be known about its
performance in less than large samples.
This paper should help to remedy the
situation a little since its main purpose is to illustrate in numerical
terms what can happen to the lower moments of a maximum likelihood estimate
for a single parameter after it has been jackknifed.
We also spend some
ti.me considering estimates of variance of maximum likelihood estimators and
their jackknifed forms. and the problem of setting confidence intervals for
the single parameter.
The only prior publication on the jackknife that seems
to be directly relevant to our study is that of Brillinger (1964).
We have elected to nlake comparisons of maximum likelihood estimates
with their jackknifed fOl;ms via the t:'ornents of their sampling distributi.ons.
but recognize
the limitations that this approach imposes.
The exact
moments of these distributions are nearly always impossible to derive.
this is so we use two lines of attaC'.k;
When
firstly. we approximate them by the
first few terms of a series expansion and secondly. we calculate them from
computer simulations of the problem.
Using these simultaneously (or the
exact moments instead of simulation if they are available) also gives an
accurate picture of the snmple size needed to validate the series expansion
in practice.
Let us be more specific about our objectives.
The jackKnife is a
procedure that hes been shmm to have certain desirable properties for a Hide
range of problems and been conjectured to have many more.
resolve some of these conjectures we will be
addressin~
In order to help
ourselves here to
questions about jackknifed maximum likelihood estimators like the follmving:
(i)
(ii)
how much bias does the jackknife eliminate. i! any?
does jackknifi.ng reduce variance and if not. is its net effect on
mean square error favourable or not?
- 2 (iii)
does the jackknife cut down the third moment of the sampling
distribution, and what effect does it have on the fourth?
(iv)
how does Tukey's well-known estimate of the variance of the
jackknifed estimate compare with the standard one and how
satisfactory is it as an estimate of the variClnce of the
..
maximum likelihood estimate itself?
(v)
does the distribution of the standardised jackknife estimate
tend more quickly to normality than that of the maximum
likelihood estimate?
(vi)
which estimate, maximum likelihood or its jackknifed form,
leads to the better confidence interval, and how do they compare
with Tukey's method for setting confidence limits?
since the jackknife is a procedure with high l;>pplied potential it
scems to us that questi.ons like these ought to be resolved.
Simulation of the problems in this paper have presented a few challenges
that h:we been overcome.
Second-order moment formulae in
principll~
should
have been a minor problem, since almost all the ones we need aI'e summadsed
in Bd llinger' s paper.
Unfortunately, however, there do seem to be a few
minor typing errors in that paper and we have thought i. t worthwhille recording
here what we believe to be accurate versions
ot these
fo~~ulae.
In particular,
these are needed if some of our recommendations are to be adopted in prClctice.
2.
Notation
We shall suppose that we have a random sample of
si~e
n, Xl' X , .•• X ,
2
n
from a single-parameter distribution with density function f(x; 0). This
function will be assumed to satisfy certain regularity conditions that will
become apparent as they are used.
To add to the
Ben~rality
the basic random
variable is taken to be vector-valued, but since the form of the algebraic
n
- 3 -
results is the same whether X has one component or ten, we will not hother
to use the conventional bold-type lettering.
In order to condense the
algebra of the next section we will denote
f(x.;
1
e) by f.
1
[a g f i )
-E -DZ--·
by I
E (a3l0~. f i )
by J
E[~~~g
fi ]
by K
E[~lO~!.]
by L
2
10
ae
ae 3
ae 4
aet.
1~~S-)
ae
n
by a. and
1
l
r,og
f,
~_. _ _
1
l: a
i"l
)
+ I . hy b. and
1
af)2
)
{:'l~,:;
Jt
by c. arid
K}
by d. and
r':.".·s t}
by e. and
ae 3
)
{'"og f;
ae 4
ae S
by A
i
n
b. by B
};
i=l
1
n
1
c. by C
~:
i=l
1
n
1
d. by D
L
i=l
1
n
1
(1)
e. by E
l:
i=l
1
Evidently all of the b's, c's, d's and e's have zero means;
f(x; e) is such that E(a.) .. 0 also.
1
l~e
assume that
In addition, \.e shall denote
(1,11)·
(.a_:~_:_~!.._~)
} by
(1, 1, 11, 11, Ill)
(2)
and so on.
This is simply the single-parameter version of a multi-parameter
notation used in Fryer & Robertson (1972).
- 4 -
In this paper, the maximum likelihood estimator of e, a , is taken
n a log f.
to satisfy th e equa t
'"
.. 0' As far as the J'ackknife is
10n
• t..
ao 1 A
la-a
1=1
concerned, we shall suppose that the basic data is divided (at random if
r < n) into r separate, equal-sized groups with n '" rs.
a when
estimate of
The
the i th group is omitted is denoted e(i)'
notation the jackknifed estimate, 0p' is then defined by
~~ximum
Using this
eAp .. {A
rO
-
0
IS
1
_
r
r A
1: e • Where the epi = {rO - (r-l)e(i)}
A
likelihood
1 r A }
(1--)
1: a .
r . 1 (1)
1"
A
are the usual pseudo-values.
i=l p1
Again, in order to simplify the algebra later on we use Aj to denote1:a
i
where the summation is taken over membeu of the jth group, with a similar
meaning for
3.
and
C., D,
J
J
B"
J
E ••
J
Homent Expansions.
lve start 'i':'th the maximum likelihood estimate, a. The local Taylor
li
a log f.1
expansion of 1:
in a neighbourhood of e gives
i=l
Ie'" a
n a log f.
(B - n1) + I (8 - 0)2 (C .,. nJ)
.. 0 .. A + (0 - 6)
. r ------~
ao
--ae--
A
A
A
A
lO=6
1=1
.,. ~ (6 - 6)3
(D .,. nK) .,. ~4
(6 - e)1I
(E .,. nL) .,. •••
(3)
Inverting (3) about the origin, we find that
(0 - 0) ..
where
Cl ..
L .,. 0 [An55]
!:...,. ! .,. L +
n °n2 n3
nil
A
I
II .. AB .,. A2J
12
21 3
(4)
..
- 5 -
(5)
.
2r
r
2r+l
In general E(A ) is O(n) and is of the same order as E(A
) and
similarly for other terms.
To find series expansions for the moments of 6 about 6 we now have
to evaluate terms like E(A 2B).
All of those that are needed for the
.
order we are considering here arE! listed in Appendix 1.
Using them and
the symbol Ese to denote the series expansion of an expectation (and Vse
for a variance) we find that
Ese (6 - 6) .. niz
+
+
.
1_
[(11, 1.) +
pJ
1_ [(11, II, 1).+ !(111, 1, 1) +
n 21 3
n 2 I t<l-
[1. J(ll, 1, 1)
2
l
1
Ese
n J2(I,
n21~.'
_L
(8 - 6)2
... -!
n1
1
I,
+ n2111 [J (I,
E
se
(8 - 6)3
A
E
se
(6 -
6)1j
1
+ --!n 213
1) + 6(11,
["_2 (II, I, 1) + 3(11, 11) + 3(111, 1) +
[(1, I, 1) + 9(11, 1) +
..
+ 0(n- 3)
3
KJ
1)2 + 12J (II, 1) + 154 J2]. + 0(n- 3)
.. Iifj3
~
i J~I
I, 1) + 6J (11, 1)2 + 15 J2(ll, 1) + 15 J3.1 + 0 (n- 3)
2
8
-~
n~1
i ~1
+-6 R (1, I, 1) + 3(11, 11)(11, 1) + 3(111, 1)(11, 1)
+ 3J(11,11) + 2K(11, 1) + 3J(111, 1) +
+
(Ill, 11) + !(1111, 1) +
i J]
+ O(n-3)
(6)
These results, which cover both the discrete and continuous cases, check
with those of Bowman and Shenton (unpublished Oak Ridge National
- 6 -
Laboratory Report) which were derived for the discrete case only by a
very indirect method.
Further terms in the expansions are not a
practical proposition at the present time.
Turning now to the jackknifed estimate. 0p' we see that an expansion
A
for (0 p - 0) is easily asse~bled from (4). For the present time we will
regard the number of groups. r, as being fixed as n increases, and consider
A
other possibilities later on.
The following expansion of (9
- B)
p
is
A
enough to determine the seeond, third and fourth moments of 0 about 9 to
.
-2 •
P
order n
A
(9
P
«* +
r8*
- 9) • --- +
n
2
where
+ •••••
(r-l)n
A
«* . 1
8* •
t2
(AB -
+..L
[A2 -
213
yot.
~AjBjJ
J
!3
EA.2]
j J.
r(l + !)r AB2 -
I'
+ EA.B~
+ 1(1 +
j J J
!)
r
AEB.2 - 2BEA.B.
jJ
JJJ
A2C - AtA.C.
j J J
ICEA.2 + I EA.2C.]
j J
j J J
1 [,3·
1
2
+ -rtt
-(1 + -) JA B - 3JA EA.B.
I
2
r
j J J
-
..
1 (1 + -)
1
-3 JB
EA. 2 + -3 J EA. 2 B. + _.
2
. J
2
. J J
6
r
J
I
J
KA EA. 2 + ! K EA. 3]
j J
6 j J
(7)
- 7In all of these terms the range of summation for j is from 1 to r.
The bias in 8
p
can be found by linear operations on the bias for 8,
but to save space we omit the details, and simply quote the final
result.
To derive the moment expansions for 8 p we now need to evaluate
terms like E[A 2r.A.B:] , the maximum likelihood counterparts of which
j J J"
were given in Appendix 1.
All of those needed for the expansion (7)
are set out in Appendix 2.
Summarising, we find that
r
MZ + 0(n- 3) where M2
Z
(r-l)n
2
of the n- term in E (0 - 0).
se
E (8 - 8) =
se p
r
[(1,
E
se
(8
p
+
is the coefficient
r
(11, 11)
1, 1) + 6(11, 1) + 3J]
(8)
- 8)4
Because there are usually fc\·;er terms in the moment expansions for
than there ure in thei r maximum like lihood
counterpart~.
ep
it is tempting
to conclude that the moments of 8p are therefore smaller in absolute
terms, but this remains to be seen.
In these expansions we have assumed that the number of groups, r,
is fixed.
However, some might argue that it is more natural to fix s,
the nunmer of observations in each group, and let r
~ ~
as n
~~.
For
this reason we have also traced the algebraic steps holding s constant.
- 8 -
It turns out that the results for fixed s to order n-2 can be very
simply obtained from those for fixed r.
All we need to do is to set
r = n in (8) and retain terms in n- l and n- 2
simply omit the factor (r
2rr
Equivalently we can
in the expansions at (8).
Only if we went
on to consider terms of order n- 3 would fixed s show itself clearly.
Of course we are not restricted to fixing r or s as n increases either we could take them both to be proportional to n l for example.
However,
formulae in this case are·much more difficult to derive and we will not
attempt to deal with this case here.
It j.s probably worth mentioning here that several checks that we
have applied to the jackknife results at (8) have turned out to support
them.
For instance in the case of a Poisson distribution with mean 6
when both 6 and
are the sample mean, the appropriate terms in the
6
p
moment expansions of
a
p
tha~
sum to zero as they ought, despite the fact
terms like (II, 11) and (Ill, 1) are non-zero.
We also evaluated the
moment expansions in a couple of very simple cases where we knew that the
second order terms did not sum to zero. Estimating a in the negativE:;
-ax
exponential density, f(x) = ae
, x > 0, gave for example
E
se
(a
a2
p
2r
a2
- a)2 =-+
n
n 2 (r-l)
+ •••••
(9)
compared with the moment of the maximum likelihood estimate
E
se
~
62
(6 - a)2 = -n
+
(10)
Again, estimating a when the data come from an N(a 2 , 1) distribution
leads to
E
se (ap - 6)
=
1__ +
r
2
4na
32(r-l)a 6n 2
+ •••••
(11)
compared with
E
se
(6 _ a)2
=
1_
4n6 2
+ .......;1;.;;5
_
+
(12)
- 9 -
Since the expansion for the variance of 0 in the first of these examples
is
(~~ + ::~)
and (_1_
4n0 2
+
_-lL
+ ...
64n 2 0 6
J
in the second, these are
both cases where in the series sense jackknifing reduces both the bias and
the variance of the estimate (r '" 2 in the first case actually gives
equal v.;triance).
This also seems to be the appropriate place to record moment
op • The standard
expansions for estimates of the variance of 0 and
procedure for e~l:imati.1g the asympt;otic variance of
~I and substitute
0 for 6 •
is to take the form
v.'e denot!> this stad stic by VI and this also
~
serves as an estimate of
0
"
vee p ). Substituting
Vee) and vee p ) which
.
1
0p for e 1n:oy gives an
~
alternative estimate for
we denote by
concentrate her.e on these two estimates together with
r
estimate of v.dance, S2 =
1) I: (0 • T r(r-1
i"l pi
~ ~ 8(i) -
special circumstances.
(r-~
we1J-known
There ar.e other estimates
P
r(r-T)"
re 2), but thesE:'. only' seem likely to work \,'",11 in
i=l
r
Tuk~y's
1
which might be used, for examp'le,
and J!-l)_
e )2.
V2 •
p
D.R. Brillinger (1966) has suggested the use of
r
(L
02(i)
ra 2 ) in certain situations where the bias of a is small.
i=l
As far as maximum likelihuod estimates are concerned ,,,e can. ShiN that
r
E
r (r-l)
se l.--r1
where - nI2
r
(r.
i=l
e~i)
J".I
[0.1, 1) +2
~
1
w_
[(11, 1) +
nX
I]
+ O(n
-2
)
(13)
is the first order term in the bias of 0, and this
of course squares with his suggestion.
Using
E
se
(VI)
expansions for
=·L+L
{L
r!+
nI
n
1 [2
2
3
e
and
(111
0p
'
given previously we find for VI that
1) + (11, 11)
-T---
+
(11,1,1).1
-
2
(4)
- 10 -
E
A
se
(V
1
-
1 )2
1)2 + 2J(11. 1) +(11, 1)2J
nI
As far as V2
1
is concerned Ese (V2 - __)2
nl
A
Ese (Vi) =L+L
nI
n2
{ 1.
13
.. E (V
se 1
+ O(n- 4 )
(15)
to order n-3 , but
- 1__ )2
nI
11) + (11. 1,
[!2 + (111. 1) + (11,
1)J
--r--2
(16 )
The moments of 5 2 are a different story, however.
T
A straightforward
but very tedious piece of algebra for fixed r produces the result
+ __.I..-n 2 (r-1)I 4
[21(11. 1. 1) + (1, 1, 1) J + 21(11, 11) + 4(11, 1)2
+ 31(111, 1) + 9(11, 1) J + IK + 3 J~+ 0(n- 3)
(17)
.
r(2r-3)
.
r
and replaclng ----- by 2 and - - - by 1 in this expression gives the
(r-1)2
(r-1)
(5 2
E
T
se
-
+ 0(n- 3), giving it zero estimating efficiency
L )2 =
2
nI
n 2 1 2 (r-1)
A
in the limit compared with V and V • Using fixed s on the other hand
2
1
]
-3
reduces E (52 - ~_)2 to order. n which presumably adds to the case for
se T
nI
using fixed size blocks.
~stimating
Incidentally, several simple checks (for
1p1e
eX811
the mean of a Poisson distribution) applied to the formulae at
(14) to (17) again proved to be positive.
Finally in this section a word about estimating the variance of e
unbiasedly to second order.
e
for
In view of (14) we cannot simply substitute
e in
E
{ se
(0 -
e)2 -
r-<11.
1) +
JOJ2)
2"
f
where E (e -e)2 is given at (6).
se
~
11 -
.
We have also to subtract the second order term in Ese(V l ) and when this
is done an unbiased estimate becomes
{
~l'-' ' 21 1
n 21 3
+
where
of
[~+ 2(111.
+ _1_
.n
1) +
~
(11. 11)
[J(l. 1. 1) + 3(11. 1)2 +
1_
n 21 4
...
e is substituted for
.
..
e.
+l
J~
(11. 1.
1)']
J(ll, 1) + 2J2]}
(18)
Similar remarks apply to the estimation
Vee) using ep and tC" Vee p ) using either ep or o. 1-Je could also consider
es timating
e i taelf along the same lines. sinc.e on substituting
in
{8 + n~2 I: (11.
4.
~ome
1) +
~
]} we get an unbiased estimator of
6
for
e
e to order n-1.
Illustrations
We now turr tc Borne concret.e estimation pr...;'lems (which· were selected
~ore
for ease of computation than for their intrinsic interest) to show
how things can work out ...umerically in practice.
Before 100Hng at the
details the-I1gh, we ought to make it clear from the outset that ,,,hilt 'Ne are
doing is
con~aring
the
certain situations.
It
perfo~lence
~oes
of the
t~o
estimators, 8 and
.
8 , in
p
not follow that we would recommend the use of
illper of thflm in any particular instance, since
8.
completely di fferent
kind of estimate may be clearly preferable.
~xamp~e
1.
We suppose that X has the special beta density
f(x) • (8 + 1) (6 • 2) xe(l - x)
and that we want to estimate
formulae for 6 and
alog f
ae
I
•
-a 2
{
and
8 > - 1
p we need the following facts
1
•
<
1
+ (2+6)
1
+ log x
I}
(1+e)2 + (2+8)2
<
1
In order to calculate the moment
6
• (1+8)
10,8 f,
ae 2
6.
x
where 0
(19)
-12-
L
= 24{-_1__
(1+0)
-S + __(2+0)
l_-S}
= 0,
(II, i) = (Ill, 1)
From this we can deduce that
(II, I, 1) = -1 2 and so on.
ShOl~ that t
It is qui te simple to
a single sufficient statistic for
e
n
1:
= 1
n
i=l
log x. = log. G say is
1
and that the basic density is a member
of the well-known Koopman exponential class.
The maximum likelihood
estimator for 6 is given by
(20)
and this is also sufficient.
Clearly jackknifing here will destroy the sufficiency
property, but this may not be such an absurd thing to do if we cannot find a oneto-one function 0f log G ",ith suitable bias ana variance.
have said before
,~e
In any event as He
are using this estimation problem only for illustrative purposes.
First let us look at: the results for bias which are given in Table 1 for
selected va:ues of nand
O.
~~ote
that in the series vC\lues the contrihution of
.
-1
tlw term 1n n
completely dominates the bias of
jackknifed estimate is virtua)ly unbiased.
0, and that as a consequenc<' the
He have thought it sensible to si.mulate
the problem too in order to check on the accuracy of the series results.
e given
simulated results for
The
in Table I appear to be pretty much in line ",ith the
'i'eri.es results, though a very large number of generated samples is needed to
establish this.
For obvious reasons even Dlore simulation runs are needed to check
on the accuracy of the bias of
iable than those for 0
beca~se
e.
p
Although the results we have here are less rel-
of the prohibitive number of runs needed, they do
confirm the order of magni tude indic.ated b)' the series formula.
In all cases the
difference between the series and simulated values is covered by two standard errors
of the simulated estimate.
All of the results in Table 1 assume that r is fixed and
set equal to the prevailing sample size and this is the optimal choice for r.
Usin;
the series formulae for fixed s ('" 1) gives very similar results but they tend to be
slir,htly further
al·ray
from the simulated values than those for fixed r = n.
-13The corresponding series Dnd simulated calculations fer the
variances and mean squared errors of the estimates ore also set out
in Table 1.
Note that in the sed es results the firs t: order term is
dominant unless the sample sbe is very small and that jackknifinp,
which always reduces variance, leads to
error in the smaller sized samples.
ust.~ful
gains in meen squBl"ed
All of the simulated values
sug~est
that the series results in Table I are close to the true values of the
second order moments.
We have BRain quoted results for fixed r = n.
Calculations of both series and simulated second moments for other
values of
l'
indicate that
and this is what
",e
We
should ah'ays use as many groups as possible,
would expect to happen in general.
The series results for the third moment that we have derived only
to order n
-2
E
give the ratio
(0.'0) 3
se ...
_._-_
E
se
~
-
a value of 7 1 ,
The simuletionc.
4
(0 -0)3
p
however, give somclvhat different ratios ,,,hen n is small and really qui te
diff~rent
values for the i.ndividual third moments from the sedes
results when n
<
20, es ",e can see frem 'fable 2.
entirely unexpected.
This of course is not
a
\-lhether the measur.es of skewness for
Rnd
similar or not depends on whether the moments are taken about
respective means.
Taken about E(&) or E(e ) the
p
especially when n > 6, but the picture if; very
the location, because of the bias in
e.
m~asures
diffen·~t
a
p
an"
e or about their
are very similar
if
(3
is used as
The results for the fourth
moment displayed in Table 3 show the same trend but are less pronounc:ed.
Judging by the calculations in these tables for the third and fourth
cumulants the distribution of 0
p
normal density about
distribution of O.
a
can be adequately approximated by a
for much lowe l' values of n than can the
Again in both tables fixed
l'
sample size since this produces the best results.
s = I give very similar values.
is set equal to the
Results for fixed
-14-
MovinE on now to estimates of the variance of 6 and
6
p'
we
find some numerical values for expected values set out in Table 4.
-2
The simulated results indicate that the series values to order n
adequately represent the moment in question.
and set equal to n.
Note that E(V2)
<
E(VI)
for series and for simulated calculations.
In all cases r is fixed
E(Si) invariably both
<
Co~paring
the results in
Table 4 with those in Table 1 we see that VI gives the best estimate of
~
V(a)
for both the geries and simulation results.
si
appears to have least
~
bias for estimating E(6 - 6)2 using the series results, but comparable
bias with Vl when the simulated results are contrasted.
standard error of Si
is roughly double that of VI and this ratio goes
Vl has slightly less bias than V2
up rapidly as r is decreased.
(which is always closest to ni ) as an est~~ar-e of E(a
slightly larger standard error.
here for using
V1
However, the
p
-
6)2 but
There is clearly a case to be made
whichever of the three measures
Wi'
wish to estimate.
Finally for this illustration we :;ive some numerical comparisons for
confidence intervals.
We are mainly concerned here with comparing the
probabilities of covering the true value of
methods are used.
Method I uses the nominal 100(1-a)% level interval
, whe!"e
normal density.
e when three approximate
Hethod
a
is the upper 100 2" % point of the standardised
similar and based on
ep -+ ~CY1
72
Hethod IlIon the other hand uses 6
+
p-
t aT> (r-l)
number of groups used in the calculations and t
a
~
.rv:
where r is the
is the upper 100
2"' (r-l)
point of the central t distribution based on (r-l) degrees of freedom.
There are other variations that we might have tried like
r;:-'
~
or
[a
.:!:. ~a
Y
v2]
[eP-%
+ ~
~l]
-
(or even modifying the number of st<l.ndard errors used)
2"
but the line had to
2a %
-15,-
be drm·;n
somet~here.
Estimated c("Jnfidc\lce c.oefficients (,dCh standard
errors of generally less than
set aut in Table 5.
1J %)
for various values of 0 and n are
Method I seems to
intervals (given only for
11 ..
eive
sli~htly too many 'low'
20) but is othend.se satisfactory.
Hethod II i.s poor for smaU samples but improve.s cw 11 increases.
For
n .. 10 the confidence coerficient is some 3-4% too low and there are
far too many 'low' intervals.
In view of the fact that
ep
is virtual!y
unbiased and the comparati ve rest~1ts for Sketmefls and kurtosis this is
something of a surprise.
It is true that V2
is biased downwards for
V(fJ ) and so replacing it by V1 \"hich is biased tlp\mrds mllY
p
improve the ovetallcoefficie.nt.
were
l"(~gistered
nut even if all of the improvement
in the '1m·.. ' interval area, it
'vouid still produce a
use fixed r .. n
bl~ttcr
,,'ell
balance.
seems
likely thllt Nethod r
All of the results for Hethod II
other values of r mere'y make thingr-
WOl:'SC.
Method III seems to give satisfactory overall coverage but the split
between too 'hieh' and too 'low' intervals is won:;'i!:g.
again gives the best results.
coeffi.cient,
il1t~r.va1.s
Fixed r ... n
In term:, of physi.cal length fOl: a nOI:111::.11
generated by Hcthod I ar.:- some"'hat longer than
those for Method II but consi derably shorter than those pro(!llced by
Hethou III even
,~hen
n .. 50.
But this is only to be c,:pectecl in vie'"
of our previous results.
pI q
Our second illustration concerns the estimatioll cf 'i' ~
(rather than the more usual lof, odds) t"here p i.s the probabU:: ty
success in
a llernoulli.
trial and q ... 1. - p.
In terms of
<1.
of
the frequency
function of the number of successes, X, in a single trial is simply
!pX
f(x; cjJ) .. Ti+~T
for x .. 0, I .
(21 )
-16-
The maximum likelihood estimate of ¢ in n trials is given by
A
cj>.=
o/r
for k = 0, I, 2 ••• (n-1)
TF-Tr)
where k is the total number
n
When k = n there is no maximum likelihood estimate, but
of successes.
if we define
cj>
= c kin here where c > (n-l), then viewed as a whole
¢
is a one-to-one function of kin' which is a single sufficient statistic
for
9, and is therefore a sufficient statistic for ¢ itself.
obvious problem with
value for c.
in
cj>
pr~ctise
is that we have to nominate a suitable
To avoid generating several sets of results for different
values of c we have decided here to make comparisons between
jackknifed estimate,
distribution in the
cj>
p
,
upp~r
appr~aches
and the
cj>,
using a truncation of the basic binomial
tail.
~~en
the truncation is light,
is not too large, n is not minute and c is not outrageous the
from the two
The
cj>
co~clusions
will be similar.
Of course we have the same kind of definitional problems with ¢ •
p
As we reduce the
valu~
of r so this difficulty becomes more acute and
the further we have to truncate the underlying distribution to avoid it.
But the higher the degree of truncation the wider the gap becomes between
the properties of the Lppropriately defined estimate for the full
A
distribution and those of ¢ over the truncated distribution.
p
At the
same time ¢ becomes less and less a maximum likelihood I"stimate (the
subject of this study).
For these reasons we use r .. n only for
simply omit the points k = n, (n:-1) from the sample space.
ep
and
This turns
out to be one of those cases where jackknifing with r .. n does not
greatly complicate the form of the estimate and it is very simple in
fact to show that ; = ; [ 1 - (n-ll.p
n(n-k-l)
J
for k .. 0, I, 2 ••• (n-2).
A
Note that 0 < ¢p -< ¢ .
By suitably defining ¢ for k .. nand (n-l)
p
A
(and changing it at k = (n-2»
we find that ¢ is sufficient for
P
cj>
too,
-17which is not what usually happens when n sufficient statistic is
jackknifed.
~
The value of
p
at k = (n-2) needs changing for two
A
reasons.
Firstly, even though
¢p
is not continuous it can destroy
the monotonic nature of the estimate and with it the sufficiency.
Secondly, ·and more important, it can lead to an outrageous estimate
of ¢ with high probability.
In this case ¢
¢p
97
= ~.
= 49
Consider for example n
and ¢p = 0.49 though when k
= 100
= 97,
¢
and k
= 97/ 3
= 98.
and
l{hen ¢ is very large we will tend to get large values of k
so in this situation;with k =(n-2),¢p will produce a disastrous estimate.
A
In practise we might replace it by ¢ or ¢
that we comment on later.
*=
Of course using;
k
(0":"j{"":i1Y , an estimate
p
=~.n
at k
=:
(0-2)
when ¢ is small will alrr.ost certainly be a point in its favour but
the probability of observing that number of successes will then he
negligible.
Beca~se
the value taken by ¢
p
little effect on its moments unless
~
will have very
is large, we have not adjusted it
in any way here since we always take ¢
takes at (,,-2) wi 11 not affect. the
= (n-2)
at k
~
2.
mom~nts
Of
co~rse
the value that ¢
\olhen ¢ is large ei tht)r provided
that we can then p~rsuade ourselves that it is ~ that we are interested
p
T~ere is no need for simulation of the moments of
in rather than r!!
q
these estimates in this binomial example since the exact values are
easy to compute.
computed
Series approximations are ..mncc(,F.sary too but we have
them (on the asslwntion of a full binomial distribution) for
several values of nand
merely to see if .md ,,,"hen they adequately
cP
represent the exact values.
Before looking at the details of the sampling moments we briefly
.
..
.
d1scuss a thlrd estlmate
cP
*
n
= (n-k+l)·'
This estimate results from
questionning how one should choose 0 and
£
(k+£)
bias of -(n-k+6T
the bias of this estimate can
Provided that
0 >
a
p
to minimise the first order
-18-
[(l:~)
be written as
the true value of
<I>
which eliminates
with
ljl
P
{
~(l-~)
+
~}
+ 0(n-
2
)J.
Since we will not know
~
in practise the solution must be
first order bias entirely.
=1
But how does
and
'I<
<I>
E
=0
compare
?
The typical pattern of behaviour of the bias in our three estimates
is shown in figure 1.
We have used
ljl
more distinct as we raise the value of
bias for
ljl
= 0.5
of the bias in
ljl
=2
there because the graphs become
ljl.
Additional information on
and 1.0 is given in Table 6.
The initial behaviour
is peculiar and quite unexpected.
First the estimate
is biased downwards. then it passes through zero to become biased
upwards.
It then reaches a peak when n is still small and finally
tai 18 off to zero.
The series approximations on the other hand are
positive monotonic decreasing functions of n and they usually beeome
tolerably close to the exact v8lues after the initial peak in the
bias has been reached.
The behaviour of the bias in both
is quite different from that in
cp.
ljl
p
and ¢
'I<
TI."y are always biased dO\vflw3rd
and their bias tends monotonically to zero as m' increase the sample
size.
It came as something of a surprise to us to f:nd that the
bias in
ljl
'I<
is much smaller than that in ¢.
p
improvement on
ljl.
b~th
However
Putting the initial behaviour of
~
are a great
to one side it is
clear that as we raise the value of $ so the bias in the estimates takes
longer (in n) to disappear. just as we might expect.
The patterns of variance of the three estimates are similar however,
though the absolute levels are Quite diffl!rent as ""e see from figure 2
and Table 7.
First there is a rise in the region of small n, then the
variance decreases to zero after showing a single peak.
Until the first order
term becomes dominant (so to speak) which takes longer as $ increases,
"
-19-
the variance of ¢
than that of q,.
is much swaller than that of ¢ * :md vpry much smaller
p
The key question to ask now is hOV7 thinrs '\omrk out
the bias and vari ance are put tor,ether to fonn
lTlfWn
~~hen
squared error.
There
is an initial region for very small values of n where q, has the least mean
squared error (because of the behaviour of its bi.as).
of its bias ber,ins to '''ane the lead changes hand and $
But before long the smaller variance of
cf>
p
p
and
cf>
* after
* is
pref"rabJe.
outweighs the smaller bins in ¢
and the gap between the two becomes considerabJe.
way behind both ¢
1'.0 the dominance
Evidently
cf>
.;,
lags alonr.;
its early dominance, so urtless n is
very small or very large there is little to be said in its favour.
distrihution of ¢
p
converges to normality about
~)
mur:h L,ster than does
the distribution of $, if the third and four eh t::-mer,f:r. <::re to be the
criteria.
This can be clearly s£'en from Table 8 "There valncb of these
highcl' moments are shmvn before and after standardisl!don.
'fhe standard estimates of the seconl order moments of l' and
we have previously dEmoted by VI and V2
(')2l0g f)
·---;;2-
take E (
'p
that
have ver:" simple forms j f ...l e
= E {(1+¢)-2- k<P- 2} over the full binomial distribution,
Lut considerably more c.omplicated functi.ons if we take
into account.
II>
t~le
Since we takE' <b to be relatively small the
betHeetl the two for any particular
est:irr,·~te
truncation
llun~0ric:ll
will also be sma]], ,ud
have opted to scrutinise the behaviour of the simpler estimates.
diffe-rence
~o
I.e
The
moments of the Viand s~ though are tak{,n f)VeT the truncated distribution usetl
previously since otherwise ','e have the same defi.nitional problems as
before.
Very little effort shows that VI
substi tute .p for $.
~
_;(1+;)2
....._,-.. and
n
Sli.ghtly more algebra produces s~
-20-
Clearly V2< Vl (when k
t 0)
anrl Vl
<
Si
unless k is very small.
Note
S2
T
.~.-'
that
a
4 when k
= (n-2).
Because of these relationshi pro the
Vl
results in Table 9
~o1ill
come as no surprise.
tends to over-estimate
Vl
both V(q,) and V(4)p) Hhilst V2 t£mds to over-estimate V(cf>p) but underestimate V(q,).
si, on
the other hand is badly biased upward for all of
the second order moments unless n is large (due to its b£!haviour for
large
k presumably).
Furthermore the standard error of
Sf
is often
considerably larger than that of Vl and frequently several times the
In this binomial problem with the
size of the standard error of V2
choice of these three estimates we would use
V2
to estimate
V(¢ ),
P
Vl to estimate V(¢) and discard Sf altogether.
Finally, for this example, we have some coverage probabilities and
expected lengths of the approximate confidenl"c intervals referred to
previously as methods I, II and III.
We could develop exact confidence
intervals here of course but this is not the point of
The length of each interval is zero when :, = 0 but
~·]e
~he
exercise.
feel that this is
inappropriate for an interval based on the normal distribution (or t) and
so
~""e
have also conditioned out this point.
All three methods tlsull11y
g:i,ve acceptable overall coverare levels "'hen n > 20, as we can sel' from
Table la, but the split between the number of too 'high' and too '10,,,'
intervals is very bad for each interval even for n
= SO.
The accuracy
of coverage for method II Seems to worsen more rapidly than for the other
t~o1O
intervalG as
we
raise the value of
~.
This is a pi ty because its
expected length for the same coverage probability is usually shorter than
that of method I and often considerably shorter than that of method III
(c.f. Table 9, also).
This concerns the estimation of the parameter A in a pO\o1er
series distribution'which has frequency function
-A x
f (x) = _L...L_.
(l-e -A)X!
for x
= 1,
2. 3, ...
(22)
-2]where
~
>
O.
~, ~,
The maximum likelihood estimate of
is found from
the equation
-x
(1 - e
and since
-~
x is
-
)
~
=0
(23)
sufficient for ~ the same is true of
~.
There is an
extensive literature on this type of estimation problem and a considerable
amount of work has been done on this special case of estimating
~
when the
zero count in a Poisson distribution is ruled out.
The minimum variance
unbiased estimate of ' A, ~ (which is a multiple of
i), has been explored
by Tate and Goen (1958).
The multiplier for
xbasically involves
Stirlings numbers which means that it is not simple;
however Tate and Goen
provide tables for it and a recurrence relationship for it has recently
been given by Ahuja and Enneking (1972).
Ad hoc estimates of
~
have been
given by David and Johnson (1952) and Plackett (1953) among others.
best of these is Plackett's unbiased estimator
number of elements in
exact variance
[ >. +
th~
~
*
.
..
rn
1'=2
n
sample taking the value r.
>. 2 (e >. - 1)
-lJ
In,
The
r. --~ where n is the
l'
This estimate has
and has been shown to be about 97"1.
efficient against the Cramer-Rao bound when>. • 0.5, reducing to a minimum
of 95% or so at
~
= 1.355
and then increasing to 100% as >.
+
Tate and
GO.
*
1
~
1
Goen also show in their paper that nl
< Vo..) < V(>. ) where nl is the
usual Cramer-Rao bound.
for using
for
~
* if
In practise there is evidently a strong case
zero bias is required, even though it is not sufficient
L
However it is possible that the maximum likelihood estimate
~
will
nI
.
1 .
h ave variance or mean squared error below
and that Jackknifing will
~
this is one of our reasons for looking at the problem.
We have computed
and
~
*,
reduce it still further to give worthwhile savings over
and
the usual series approximations to the lower sampling moments here and
·-22-
and have also checked their accuracy via computer simulations.
All of
the results for the jackknife use fixed r ~ n and this is almost invariably
optimal.
The elements needed to compute the se....ics approxi.mations are
easily Mrived, but 11luny are quite lengthy in appearance.
For t'xmnple
.
(1 + 3>' +
(1, 1, 1) .. ··..·-····--::.r··A2 (1 - e )
+ •.
.1
.
(3e- A(1+A) - (3+A)]
+ ---_•.-.• '.-•.'----'------'-"'-
A2 )
A(l - a-A):!
._
4(1 + A)
-------:.::~--
A2(1-e ->- )3
A3 (1 -
e
)2
and so to save space will not eive the moment formulnc in detail.
Before turninr. to the numerical resul ts, hmJever. there are one or
two minor points thi:t we ought to make about these estimates.
= 1 for all i then A .. 0
ncte that if
Y..
of
calcul..tt~_tlg
;I..
Hhen
1
the
point to avoid complic:at:i.ons.
vBI~es
as well
a~
l1cnts of ),
l!l(lI
Firstly we
which is an 'illegal' estimate
'~e
have included this sample
It is also possible for \
t~
take negllti ve
zero (thourh the probability of this will usually be
very small inceed) and thi s has b(>en treated in a sit:1iJ ar '''ay.
note that Sf
=0
Finally "ye
if the means (If all the sub-groups (",hen jackknifing)
are the same, and
confidence intervals of zero length have been excluded
in the simulations.
Some series bias results for n
=5
~re
set out in Table 11.
Evidently
the first order term for A is dominant leaving the jackknife virtually
unbiased, but the overall level of bias in A is small (less than 5% of value of A)
-23-
even when n '" 5.
So although jackknifing reduces the bills by 95% or so,
the total saving is scarcely
\~orth th~
effort, and as n incre,1ses the
Being so very small the bias in A is
bias rapidly approaches zero.
p
difficult to establish by simulation, but the results that \-1e have tend_
.
to support the series level for it.
Simi.lar remarks app 1)' to A•
A
Some calculations for the second order moments of A and A when
p
n .. 5 are given in Table 12 and they are typical.
The results are
disappoi.nting, principally because of the dominance of the first onll'r
term.
The mean lIeJuared error of A is always above the Cremer-Rao
p
bound and exec.eds both the variance and mean squared error or A for
small values of ),.
By tLe time
on both the bias and variance of
E
fA -A)2
E
(A-A)2
From then on ---~!':....'-!-_..
se
A ~ 2, however, A is cutting down
p
A, thoueh by very li ttle in
0.99 as A increases.
the behaviour of A is worth commenting
s~cond
order terms in V (A) and
se
r. se ().-).)
overall level::: below the Cramer bound.
over A or
A* is some 13% at A
c
011 ,~hen
:lb50 1 ute
terms.
It does seem to us that
A is very small.
Here the
are negative, so takinr. tll"
The
0.1 when n
efficien~y
~
improvement in A
5 and may be worth the
effort (perhaps it is Horth seen-ching for the minimum mean squared er.ror
estimate in general!:).
Sim'Jlatcd results that we hnve [or thef:e second
order moments suggest that
th~
series values may be a frac::ion 10'-1.
In view of the size of the second order moments of A and
A it may
l'
be argued that there is no longer Bny point in continuing our study of
this problem.
I!owever, as we said at the beginning our main objective is
to compare A and Ap ' not to contrast
will comment briefly on some
t~ern
with other
~stimates,
so we
other features of our two estimates.
Series
calculations indi cate that in the smaller siz(!<! samples the thi rd moment
-24-
of Ap about A is almost always at least double the size of that of ).
and often much more.
Simulations for this third order moment which
we believe to be quite accurate show that E().
p
- ).)3
>
E(~ -
A)3
also,
but indicate that the gap is nowhere near as wide as the series
calculations suggest (usually less than double).
Standardising this
moment by dividing by the appropriate power of the mean squared error
leaves these conclusions virtually unchanged.
It may be worth noting that
these third order moments are very much more in line when taken about the
respective means of the estimates.
to order n
··2
Series for the fourth order moments
are identical of course.
The simulations that we have run
here suggest that the leuding term in the series is a very good approximation
to the true value of the momen~.
Standardising them (using
3 ] typically) shows that judging by the size of the fourth
cumulant: the distri1>ution of A converges much faster to normality than
p
does that of A and likewise if the moments are taken about the means of
the estimates, but we must remember that for third order moments we came
to the opposite conclusion.
We have also run out a few ;;i.mulations on the distributions of VI,
V2
and s~.
The tentative conclusion from these seems to be that E(Vl)
~
<
E(V 2 ) < E(S~)
and that all three are biased dm''T\wards ~enerally though
not usually by much.
Also, although s~ has the least bias, as usual it h~s
the largest standard error, it frequently being ten times larger than that
of VI or V2 when n is small.
This makes the mean squared error of
si
several times the size of that of VI or V2 es a rule.
Finally a brief look at the approximate confidence intervals.
Overall coverage probabilities are generally quite accurate for all three
methods, though method I does seem to he slightly inferior to the other
two most of the time.
For example with A .. 2 and n
= 10
coverages for
•
-25method I are 82.4% (nominal 807,), 90.52 (90%) and
92.00;~
(95%).
Method lIon the other hand gives 79.80% (BO%) , 89.847 (90%) and
9/~.10%
(95%) and method III 78.16% (80%),89.96% (90%) :mc1 94.46% (95%).
The tt)il split is less satisfactory, hOI,ever, and \,hen n = 30 and A '" 2
with nominal confidence level 90%, method I splits the too low and too
high intervals as (7.40%, 3.72%), method II is (7.40%, 3.92%) and
method III as (6.34%,4.00%).
Because E(S,f> > E(VZ} > E(Vl) the
expected length of intervals for method III is almost certain to be
larger than for the other two for a fix.:'o confidence level.
n
= 10
and a nominal confidence level of 90%
IVl'
with A
... 2,
foupd the expected lengths
for the three methods to be 1.6303 (I), 1.6360 (II) and 1.7871 (III).
Raising n to 30 produced 0.9495 (I), 0.9505 (II) end 0.9761 (III).
Most of this \York was carried out \olhilst: \'le were visitors at
the Biostatistics Department, Cha!)el Hill, :lorth CaroliniI.
";e would
like to express our thanks here for support from Institute of Generol
Hedical Sciencei; Grant G.M.-12868 \·]hilst: we \-lere there.
II' addition
one of us (J.G. Fryer) would like to thank the Fels Research Institllte,
Yellow Springs, Ohio for SUPP01"t during the S\.lmtT,er I)f 1972.
his \·1Ork was carried out whilst
lIe
Part of
was a Vifliting FellOl~ at that Institute.
Refel"eneE'S
Ahuja, J.e., and Enneking, E.A. (1972)
Recun'ence relllti.on for minimum
variance unbiased estimator of a parameter of
Bri1linger, D.R. (1964).
R
left-truncated
The asymptotic behaviour of Tu!:ey'll general
method of setting approximate confidence limits (the jackknife)
w'hen applied to maximum like lihood (>s timates.
•
Brillinger, D.R. (1966)
The application of the jackknife to the
analysis of sample surveys • .~~~e~!?!y, ~ 74-80.
David, F.N., and Johnson, N.L. (1952)
Fryer, J.G., and
Ro~ertson,
C.A. (1972)
'I'he truncated poisson.
A comparison of some rnethods
for estimating mixed normal distributions.
Plackett, R.I,. (1953)
~.i_?.!~.(~~_~i:.l5:.a,
12.,
639-48
The truncated poisson distrihution.
Tate, R.F., .md Goen, R.L. (1958)
Ninimum variance unbiased estimation
~Y-.~E.'!~2:._.!:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
E(A2) " nl
E(AB) " nOl, 1)
E(A3) .. nO, 1, 1)
E(A 2B) .. n [(11, 1, 1) + 1 2]
E(Al\2) .. n [(11, 11, 1) + 21(11,
E(A 2 C) .. n [(111, 1, 1) -
I»)
IJ]
E(A4) " n(l, 1, I, 1) + 3n(n-l)1 2
E(A3B) "n[(l1, 1, 1, 1) + 1(1, I, 1)]
+ 3n(n-l)l(l1, 1)
E(A 3e) .. n [(111, I, 1, 1) - J(1, 1,
+ 3n(n"1) 1(111, 1)
1)]
J + n(n-l)
E(A2B2)" n [(11, 11, 1, 1) + 21(11, 1, 1) + 1 3
E(A3D)" n [(1111, 1, 1, 1) - K(1, 1, 1»)
[1(11, 11) + 2(11, 1)2- 1 3]
+ 3n(n'-I) 1(1111, 1)
E(AB3)" n [(11, 11, 11, 1) + ,1(11, 11, 1) + 312(il, 1)]
+ 3n(n-l) [(11, 11)(11, 1) - 1 2 (11, l)J
(xiii)
E(A2BC)" n [(111, lI, 1, 1) + 1(111, 1, 1) .. J(ll, I, 1) - 12 J]
+ n(n-1) [2(11, 1)(111, 1) + 1(111, 11) +
These formulae r.ce easily elltablished.
E (A2 BC) " E {[1: a?
i
+
l:
i,j
i~
1
r2..(J
Cpnsider, for exa",plE',
a. a. ] [1: b. c. +
1
J
j 1 1
1:
i,j
i~
bi
C•
J
.!}
•
Because the variables a., b. and .::. have zero means \1e can wri te t:Jis as
1:1,
E(A 2nC) " E
r
" E
l~
[1:
i
a?
1
(~-~.~.:-!-i
b. c. +
1
1
r
2 g
[a 1o
as 2
1
l:
a? b. c. + 2
1
i,j
i+j
~.t
+
J
r]
J
1:
i,j
ifj
a. b. a.
1
1
(.a310g_~1 - Jl
asS
J
J
C
j
]
which reduces to
n [(111, 11, 1, 1) + 1(111, .1, 1) - JOl, 1, 1) - r 2JJ
+ n(n-l) I
[(Ill, 11) + IJ ] + 2n(n-1) (11, 1) (111, 1)
(i)
= nl
E(I:A.2)
j J
E(A rA.B.) = n [(11. 1. 1) + 1 2]
(ii)
j J J
E(A.I: A.2) = n(l, 1. 1)
(iii)
j
J
E(AB.I:A.B.) = E(tA.R.)2 = n [(11. 11. 1. 1) + 21(11. I, 1) + (8-1)1(11, 11)
(iv)
j J J
j J J
+ 2(8-1) (11, 1)2 .. (8-2)13]
+ n(n-8) (11. 1)2
E(A.B.I:A. 2) . E(A2EA.B.)
(v)
j J
It
j J J
= E(rA?EA.B.)
j J j J J
.. n [01. 1. 1. 1) + 1(1. 1. 1) + 3(11-1) 1(11. 1)1
• n(0-8) 1(11. 1)
E(A.r.A~). E(EA~)2
(vi)
j J
j J
= n [(I, 1. I, 1) + 3(8-1)1 2] + n(n-8)1 2
E(A2.EB~) =
j J
(vii)
n [(Ii. II, I, 1) + 21(11, 1. 1) + (6-1)1(11, 11)
+ 2(s-1) (11. 1)2 - (8-2)1 3]
. + o(n-s) l.'-1(11 • 11) - 1 3]
E(A.EA.B~) = n [(11. 11. 1. 1) + 21(11, 1, 1)
• J J
J
+ (5-1)1(11, 11) + 2(s-l) (11. 1)2
(viii)
(ix)
(:;)
(xi)
.
- (s-2)1 31
E(AEA?) = n [(1, 1, I, 1) + 3(s-l) 1 2]
j J
E(AEA.2B.)
j J
=
T1
J
E(A 2 EA.C.) =
j J J
[(11. 1. 1, 1) + 1(1, 1. 1) + 3(s-l) 1(11,
I»)
E(AC r.A~)
j J
.. n [011, 1, I, 1) .. J O. 1, 1) + 3(8-1.) 1(111, I)J
+ n(n-8) 1(111, 1)
(xii)
E(A r A. 2C.) = n [011. 1. I, 1) - JO. 1. 1) + 3(s-1) lOll. 1)]
j
J
J
Again. only simple algebra i8 needed to evaluate thesernyments.
For instance
E(ABEA.2) = rE(A.3B.) + r(r-l) E(A. 2)E(A. B.)
jJ
J.1
J
J
J
and terms like E(A.3B.) are available from Appendix 1 with n replaced by 8.
J
J
•
~
e
e
e
Table 1
Some Numerical Comparisons of the Series Representations for the First Two Moments
Moment
A
Value
of 6
Sample
size (n)
A
A
E (ft -6) to
se
-1
order n
Ese (6 - 6) to
-2
order n
E (6 - 6) to
se p
-2
order n
Var
se
& Var
"
(6)
A
A
(6 p )
se
-1
to order n
(6 -
Var (6) to
se
_?
order n •
E
6ho
se
-2
order n
E (6 - 6)2 to
se p
-2
order n
0.2011 (0.2153)
0.1044 (0.1077)
0.0461(0.0463)
0.1679(0.1822)
0.0915(0.0938)
0.0427(0.0427)
0
2
6
10
20
0.1200
0.0720
0.0360
0.1298(0.1247) -0.0117 (-0.0207)
0.0755(0.0745) -0.0039(-0.0048)
0.0369(0.0377) -0.0009(+0.0001)
0.1333
0.0800
0.0400
0.1867(0.1997)
0.0992(0.1022)
0.0448(0.0449)
SO
0.0144
0.0145(0.0146) -0.0001(+0.0002)
0.0160
O.01tiS(O.0166)
o.Ol?() (0 .0168)
6
0.3150(0.3248)
0.1833(0.1884)
0.0895(0.0869)
0.0353(0.0386)
-0.0286(-o.C267)
20
50
0.2912
0.1747
0.0874
0.0349
-0.0095(-0.0097)
-0.0023(-0.0052)
-0.0004(+0.0003)
0.9600
0.5760
0.2880
0.1152
1. 2903 (1. 3993)
0.6949(0.7357)
0.3177(0.3261)
0.1200(0.1219)
1. 3751(1.5048)
0.7255 (0.7712)
0.3254 (0.3337)
0.1212(0.1234)
1.1635(1.2515)
0.6438(0.6750)
0.3041 (0.3099)
0.1177(0.1195)
6
10
20
50
0.4582
0.2749
0.1375
0.0550
0.4961(0.4918)
0.2886(0.2737)
0.1409(0.1472)
0.0555(0.0475)
-0.0455(-0.0611)
-0.0152(-0.0347)
-0.0036(+0.0032)
-0.0006(-0.0008)
2.4590
1.4754
0.7377
0.2951
3.2891(3.5421)
1. 7742(1.8231)
0.8124(0.8291)
0.3070(0.3073)
3.4990(3.7839)
1.8498(1.8981)
0.8313(0.8508)
0.3101(0.3095)
2.9629(3.1597)
1.6434 (1.6741)
0.7775(0.7899)
0.3013(0.3016)
10
0.0164(0.0163)
4
I
Note:
The figures in parenthese.s are the simulated values.
r is fixed and set equal to n in all cases.
As far as 6
p
is concerned
Table 2
Values of the Third Moment and M
~-~
~---
--
-J--~.~--"J
--_._--~~-,_.
-,~"~--,,,.,
0.1120
0.2194
0.0640
0.0967
2.1966
1.243..
10
0.0403
0.0580
0.0230
0.0292
1.6410
l.0151
0.0101
0.0124
0.0016
0.0017
0.0058
0.0068
1.2458
0.0009
0.0010
0.7847
1.9569
3.6324
0.7045
1.0983
0.1761
0.2249
1.1182
1.6674
I
I
I
50
6
!I
10
2
20
0.0282
0.0331
50
0.l,026
0.5663
I
\.._.
I
6
10
20
Note:
0.4056
0.5393
0.0649
o
The first entry for E(e -
similarly for E(e
p
_e)3.
rcrn:lrks apply to Tnbl~ 3.
-- ---
4.5')70
6.6860
1.6225
2.1100
0.1136
0.1164
50
0.1006
0.1231
0.0161
0.0195
---~.~..
7.8872
14.7205
2.8394
4.0975
0.7099
0.9355
4
I
O/;~,
I
0.7721
I
l'
0.4613
•
X
[E(6 p -
E(ep»~l/i
1.5994
1.3937
1.0644
1.0629
0.7660
0.7701
0.4533
0.4571
1.3500
1.2636
1.1910
1.6215
1.0212
1.0706
1.0570
1.1666
0.7135
0.7473
0.7418
0.4454
0.4442
0.7641
..
_.~.-
0.4711
._.'
--.-.~--_
...._*-..._.,-
-
.__
.. .,--
..-
-
1.9999
1.1904
1.4065
1.2958
1.5670
0.9741
1.0480
1.0558
1.1921
0.7682
0.7500
0.7577
0.6760
0.38LO
0.4259
0.4283
e)3 is the serie::. result to order n--
2
e
-
and the second gives the simulated value;
ar~ calculated from
In both of t.h('aSf~ tR~)les r i~ fixc-cl and s~t 1:'C'!u.al
(
.
E(e p - E(6 p»3
1.9678
The quantities in the other fou= columns
e
.
I
~
20
E(6 - E(6»3
[E(8 - E(e»2] 2
%
P
6
I
I
E(e - e)3
• p
3
lE(e - e)2}
E(6 - e)3
E(e - e)3 E(e - e)3
p
[E(6 - e)~'j%
(n)
0
Moment or Measure
-
---
Sample
Size
.
Value
of e
.-
.... --.__
t()
Lhe simulations.
t.htJ" s:Hr:n~,c gi~e: in
0:'1("'-,
and
Similar
r.nfd~ ..
e
f
0(
e
e
e
Table 3
Vah~e~
I
II Sa~p1e
I
I Value
of
5lZe
e
I
I Cn)
!
ii
i,
j
O!
:
!
I
I
1·
,
J
!
I
I
__
!
20
,.
I
I
I
,
2
0.OO~8
0.00 ... 8
~
I
I
::>0
0.0008
0.0010
:
--_..
!
2.7648
15.9551
j
6
i
I"
1
o
0.9953
3.5496
I
,
Ito
I
I
·
I
I
I
I
I
!.•-
-
6-
i
I
10
II
4
I
20
1.6326
3.3721
I
50
0.2612
0.3317
I
!• 0.007::>,
0.00l:~ i
[ECe _,)2] 2
l
p
.
'
"O", ..
I
I
3°8"
... \ ) :
I .
I
I
~
!
7.0462
1
') 9764
!
!I
0
!
! 3.6149
1
I
\
f 7.5414
I
i
i
4.6589
I
I
3.4623
II
: 5.9452
I
i',
1.6326
2.6946
I 0.2612
0.3004
i
•
---
5.8546
I
I
III
I
!
5.1085
!
I
4.1185
I
3.4570
i
I
i
I
lI
5.2095
Il
4.1225
I
3.4639
3.4465
6.3697
5.3335
-l
6.2079
I
I
1~.1740
5.9784
5.3144
3.4305
1._
f
6.7220
II
5.3458
I
i
i
I
!'
)I
;
I
.
!
!
I
I
i
i
4.1572
'1:
I
3.4335
I
J
I
\
I
6.5551
t
I
.J
!
i
i
I
I
I
II
5.4503
I
I
r
I
I
1.
:
ij
I
\I
I
i
I
'orI
I
I
t
4.1292
I
I
I
I
I
- '-'-'-"
.
I
_.--
!
A98:
!
I
8.1311
5.2636
5.3246
!
I
i
i
'
"
i
!
-•
Q!
I 14.9475
6.5305
3.4664
I
!' '0.248....
' ' 9 6 6 '; u.• -'"5"0
- ' ,I
'I'
a..
):
I
.
' .
:
I.
• • • • . • _-
I
~•
L
I
I
8.3975
I
_.~
I,9.1698
2.7H8
i
i
\! 0.99)3
- :1
!I
4.1246
I
I 0.0008
! 3.6482
i 0 .0009!
I
p
I
7.8373
!
I
I[ECe-E(e))']2 , [E ce -E C'p)) 2J 2 i
l
I '
I
4.5612
I
i
18.1403! 18.1403
107.977(
! 63.5952
6.5305
21.4J.85
I
II
'
p
I
"
. ECe p -ECe p ))4
I E(e-E(e))~
'I 0.0192
5.7844
!I 5.1412
,0.0452
i i i
=-=."",,,--==~-=,===",,'
i
: 9.1721
-- I
_,)4
I
0.0398
! .0-,
0.0550! 0.0/.92
50
=i.
!'"'.--===
Ecep -e)
II 0.0533
0.2602
C'
E 8
E(e~e)l+·
A
I
I[E(e-,-)2] 2
,~-l+
A
I
I
t
0.2488
.50 72
I
J
Moment or Measure
'
i
'
i..., , 0 ,!'
'j'
I
I'
i
I
,
·
0.0192
0.0671
I t ·
j
Ii
I
I•
I
!
0.4250
!
10
I
1
I
J
o •0 533
i
6
i
!
I
Ece-e)'
I
I
I
I
A.
'
I
I
!
.
of the Fourth !loment and MeasurE s of Kurtosis
4.3190
3 3029
! 4.2632
!
!!
I
3.3233
'
I,
--'---
4.'30;j5
3.3300
Table 4
Expected Values .._---_.
and Standard
Errors of.. _---._
Estimates
----------_
__ ..-.•.-_._---_
..
of Variance
for
....
_----~-~_._--
Selected Values of
_.-.-._---
e and n.
----,.---Moment
.~---_._._._---_._._--
Value
Sample
_o.J.__ .!!__ 2}3.e (n)
10
o
20
40
..'-t..=r
:;:.--=--
2
=::...-..... -::::=..:=:=
E{V2)
._-----_._-1-
0.0948
0.0971(0.0553)
0.0844
0.0849(0.0502)
0.1116
0.1173(0.1300)
0.0437
0.0439(0.0165)
0.0209
0.0209(0.0055)
0.0411
0.0411(0.0157)
0.0203
0.0203(0.0053)
0.0476
0.0479(0.0303)
0.0219
0.0220(0.0096)
=.:......=..-:::.=~-=:.~.=;:=-..-...="_--
0.6660
0.6728(0.3500)
0.6049
0.6009(0.3180)
20
0.3105
0.3116(0.1070)
0.2952
0.2949(0.1010)
40
0.1496
0.1500(0.0354)
0.1458
0.1460(0.0345)
~~~.::l.-::::-:';::' .. ~-:::;:;;:.;. _~-=-'"':'':::':'''''':_-':::=.-=:''::::'=-~;;:'''-=.::j:::=':-:'
=:::---:...~=.~~~-::.~
4
10
1.700l.
1.5492
20
0.7940
0.7562
40
0.3041
0.29'>0
1
--_.•.._- -----Note:
.....
I
I
I
0.7684
0.8074(0.8280) I
0.3345
0.3396(0.2090)
j 0.1561(0.0599)
0.1554
~--:.-
.
.::.:.:..."""'::--:~~~~.;;..-::..:;:::;=::.z.;.:--:=_~:
1.9567
J
__ ._--------_._-_._._._-_._---
.
I'
......:.=...~~::-~:::=.=,;..--:-....:: :~..=;-: =.:-.:. 'F'':'=::':'''==--=::'~-_''--'':'::=-~:':''::::'1
10
.
E(SP
0.8540
0.3133
._--------
For 0 = 0, 2 the first entrv in each cell gives the series
result to order n- 2
The ~econd is the simulated value
and the standard error of the estimate of vari 81l~e is given
in parenthesis. Only the seri es results ar.e eben when
o = 4. We set r = n in all cases.
Table 5
Method I
Value Sample
of e Size
(n)
-===---
90%
level
level
..
"S:'-
10
0
'80%
20
40
'
..
95.33
80.35
90.32
95.05
(10.47)
(5.74 )
(3.48)
90.16
95.21
Note:
90%
95%
80%
90%
95%
level
level
level
level
level
level
.._c
76.53
86.99
91.93
81.38
90.28
94.65
78.09
88.35
93.82
80.42
89.99
94.74
(15.11)
(8.93)
(5.24) (12.84)
(7.09)
(4.18)
79.47
89.26
94.16
90.04
94.83
I
-==
".
--
80.89
_._-
..!"_--- '.
10
80.97
riO.~7
95.25
77.43
87.04
92.14
81.28
90.19
94.65
20
79.40
89.77
94.82
78.14
87.87
93.49
80.06
(11. 39)
(6.35)
(3.68)
(15.10)
(9.38)
(5.44) (13.57)
89.52
(7.66)
94.43
(4.34)
90.21
95.12
80.15
89.62
94.45
90.0!1
94.95
40
-
f-Iethod III
-
80%
.-
--
::e:==:••
91.05
,-
2
95%
level
81.17
80.52
Hethod II
,--
80.63
I
80.78
The first entry in each cell gives the percentage of intervals covering
the true parameter value. The second figure given only for n = 20 is
the percentage, of intervals that fell totally below that value and so
is a guide to the symmetry of the technique.
.
Ta;,le 6
.~i~_~
r-
I
I
I.
Value
or w
in ~~ estima_te~_ c~ ...P-.ior:. selected vt.lues of nand ¢
--(1-
----
~
II
'
j ord~-
_,t~n) _~
0.5
I
!i
10
I
20
6
t.!
IIII
! 1 ·'f
,
0.1123
0.0939
It
i
I
6
!I
\20
,_ ----l 50
2.0
I
!
6
0.1667
'-0.1782
0.0900
-0.041':;
-0.0055
I~,
Exact Bias
.
l.n
¢*
order n-·
I
-0.0500
-0.0419
-0.0167
-0.0015
-0.0039
-0.0000
0.0412
0.0150
~
0.0156. --0.0007
II
-0.0006==1
-0.0000.
0.3333
0.5000
-0.5959
II
-0.2000
-0.2632
I
1: 0.2150
ij
!i
0.2000
0.2600
-0.2526
II
-0.0667
-0.0444
II
0.1195
0.1000
0.1150
-0.0299
-0.0158
-o.0002!
0.0424
-O.O_~30
-O.OOz4
-0.00~
1.0000
1.8333
-1.5837.
-1.0000
0.6000
0.9000
-1.1133
-0.3333
0.3000
0.3750
-0.2770
~
0.0298
Ii
I.
I
I
I
-l O.~.::: \. 0.04~0
11--0·5784
20
II
'I
50
;1 0.1343
10
I
0.0750
I--
.
I!
0.0419
;j
I 10
1•• 0
.
order n-
I
0.0375
!l
II
I.
I.
0.1250
II'
_.+"50~_~~:~~~~
•
I
I.
- .. I
,
~
~
Bias in ¢
Bias in ¢
"
p
;
It-I-----r-------,------j--------r-----ISamnle,'
C'"
E_X8ct
sen. es to1 Serl.es to2
10'
Serl
....
l.ze 1.il1
l:.xac t
'l
0.1329
0.3688
e
....
I
1
l
_
I
I
I
I
-0.0789
-1.01~
1
-0.4355
I
-0.0287
~~~_u~-==-~_, -0~~~_-O.0122_ _ '__~~_:~OOO
,
e
e
.-:
#
e
e
e
,
Tahle 7
Variance and mean squared error of
"
~,¢
p
and
~
*
for selected values of $ and n.
.-r--
Second Order Homents of
Value
of
Sample
Size
Exact
order n
(n)
Series to
order n -2
f
j
~
- - - - - - - - - - - p., Variance and M.S.E.
Exact
Series to
.
of 9*
-2
order n
I
!
0.2~50
-
0.1259
0.1276
0.0749
0.0767
0.1250
-
0.1213
0.1213
0.0703
0.071~7.0564
I 0.0564
0.0592
-
0.0595
0.0595
0.0225
-
0.0..24.8.
0.0228
0.0230
-
0.0230
0.0230
0.4130
0.4139
0.6667
-
1.5556
1.6667
.0.0223
0.3114
0.9333
-
0.1673
0.2366
10
0.7115
0.1577
0.4000
-
0.1200
0.7600
0.1015
0.1653
0.4889
-
0.3235
0.3254
20
0.3362
0.3505 -
0.2000
-
0.2800
0.2900
0.1800
0.1809
0.2211
-
0.0800
0.0928
0.0823
0.0833
0.0823...,
0.0154
5.4000
2.5235
-
0.0835
0.0835
0.1427
1.1725
0.1200
1.3595
2.6000
-
0.4850
0.6741
0.1875
-
0.3438
0.3594
10
0.2116
0.2204
0.1125
-
0.1688
0.1744
20
0.0752
0.0769
0.0562
-
t-~~-,,_0.?52
10.0250
6
50
1.0
50
10.0953
0~.()3.5~
I
I
I
I
~.O~~_~
I
0.2241
0.2241
-
q~O?~~
0.1143
3.0000
-
10.0000
11.0000
10
1.2711
1.2887
1.8000
-
4.0800
20
1.8908
2.0268
0.9000
-
1.5300
1.6200
0.3872
0.~639
1.0895
-
0.9161
0.9169
50
0.5013
0.5193
0.3600
-
0.4608
0.4752
0.3708
0.3111
0.3894
-
0.3939
0.3939
I
+.<?·~2,.Zl
6
2.0
Note:
Variance & M.SoE. of
0.0322
0.0640
0.2783
0.2909
I
I
-1
6
0.5
I
Series to
4>
0.37~7
4.~200
I
The first entry in each cell is the variance and the second the mean
squared error.
,
Table 8
Third and fourth moments of
¢ and
¢ and measures
of skewness and
_______________.__.
.__
__..-..P_._.
.
. kurtosis
fva1ue
I
~_.
I
~"
C ...
I·
If'
.-
0.5
! E~(jl_¢;)3
S~~e
!
I
.(n!
L
I
6
II
I
10
20
I'
-
6
I
10
. 1 0
I
.
,
I! +-
20
I
i
I
2.0
0.2956!
0.0073!
2.8569
!
!
0.0396
0.0150
t
I
I E(<j>_¢)4
I
i
1.8565
I
I
0.2923
! 0.7108
0.1588
;
!
1.1225
!
l
0.7215
I
-1.0881
0.0515
II
.
10.0028'
0.5965
i
2.1979
0.0710!
2.7497
-0.0889
I
0.5706
I 0,~41_4
6
i-0.9313
I
I
10
1.3267
I
20
7.7239
I\
I
!
!
I;
-1.3235
\
I
!
0.9225
\ 0.3403
I
!
! 0.0182
I
! 14.6330
!I' 8.7001
I
,
0.0168
I
0.0021
l
I
-4.0446
I
I
I
-1.5425
l !i
-1.7931
-0.2805
0.75~_ ~2563J
e
~.
0.9068
2.6767
2.0093
I
Ii
i·
I
;
i
I
I!
2.0473
II
0.1493
Ii
I
I
r
4.0537
4.5605
16.6637
I
!
5.5321
:
4.7259
6.5184
2.5096
'I
1.0236
,
1.3562
l
2.1508
I
-0.887643.9895
I
0.4630
I'10.7080
~.8147
,
I
I
2.1742
2.4109
e
5.7145
I:
2.5066
1.1337
!
'I'
!
l~
I
1.2411
l 4.0043
-1.1312
2.8519
i
!
I!
I
!
1.9866
,'7.0053
!
"
II
---
0.0594
\1.2806
2.3053
I
0.1768
..---.-.-
I
I
_
4.4301
4.0220
I '-1.0089
.~
i
I
' :
JI
I
I
I
!
-·-A----
. {E(,;._<p)2}2\ {E(<jl _¢)2}2
I
I
~~~1_0~~?8_=t-_~~~_!~~1_ 0:,0.:'21 b-<l.·~320
' !
oC.
•
E(ip-<jl) 4
I,
---;;-----,
I i !
'
I
-0.2523
'
E(¢-¢)4'
11
I ---il. Y
I O.OC~~ 5.2702 I
I
I
I
0.3454
.
4
E(q> -q»
p
·-:-·,:-::::.-::=--;::'f::=-=-":"'·-:'·:-::-':::·::-=·==;:=:::=:::·:':==-=:::--":":.F ---
I 1.4497 I
I
-1.4223
I
I
I i I
I
j
:
1 0.0042
1.0445
·-1==::=::·-·-::::::::::'"f====::::-·:.:::;-·-:::;:.-.:::-.:.:;:::::::::···-_· ..
!
I {E(<j> _¢)2}12
l
,
i ;
i
0.0025!
,50:
L
-0.0230 I
---~ - .--:3:/
. +--!---i--·
1.8633
50.
i
I{E(9-<P)2}~Y2
I
I
I
I
I .
I
I
I
-A---~-
!
I
50 _
11
P
_<jl)3
l
I
II
E(</!
.----l------r
!
i
I
1
0.2923
I
i
~--~-II--:(~-~~II--E~~p-~)3 -l~--A---·----~.
I .'
I
I
,
I
I.
~~le
[9.8284
I
5.9143
e
I
Table 9
Estimates of Variance and Hean Squared Error of
cjJ
and
<p
•
·-·:·-::'::-~::'=':"-===-===_::=:==--:-":··-::==··'::::::=:~:===:-::=-""'=-':;==E...---"A-·_......
Value
of ¢>
Homent
Samp Ie I - - -..---r--------,~~---___l
:ni;e
-
0.5
E(Vl)
E(V2)
E(Si)
V(cP)
and
V(</J p )
and
A
H.S.E.(</J)
p
i-.-.----+-----------I-----..:.=----+------\-------t
0.2544
(0.6569)
0.1244
(0.1279)
0.3647
(1.6963)
0.2116
0.2204
0.0749
0.0767
20
0.0795
(0.0975)
0.0658
(0.0642)
0.8281
(0.1372)
0.0752
0.0769
0.056/.
0.0564
50
0.0255
(0.0143)
0.0241
(b.0131)
0.0256
(0.0146)
0.0250
0.0252
0.0228
0.0228
_ _. '::7_7=-"'=-;-":-"_-0.."1=::::._
20
A
H.S.E. (</J )
10
10
--
1.0958
(2.0659)
0.3639
(0.8744)
......
0.2717
(0.1971)
0.2368
(0.2492)
I
-
-
2.2167
(5.9471)
0.4552
(2.3209)
0.7115
0.7577
I
0.1015
0.1653
I
0.3362
0.3505
0.1800
0.1809
0.0953.
0.0971
1.2711
1.2887
. J
2 0
2
0
2.4842
(5.7664)
0.7807
(0.6078)
5,0032
(18.6861)
1.8908
2.0268
0.3872
0.4639
0.5145
(0.6498)
0.4171
(0.3822)
0.5701
(1.1518)
0.5013
0.5193
r,.3708
50
_ _ _ _ _ _ _•
Note:
1-..
_ _._.
.. _ _._ _. . •
.. _
.._.
u.3711
..__
-----_._- .,I.
The first entry for the last t\110 columns is the variance of the
estimat.:.. and the second the mean squared error.
The figure in
A
parenthesis for the E(V ) and E(Si) columns is the standard error
i
of the estimates of \7ariance.
.
Table 10
._---.-----_._ _._..__... _--...
Value
of
Sample
Size
(n)
_..-80%
level
Method I
90%
1 eve1
Nethod II
Nethod III
8-6% ·-'9"0"%-·--9·5·7...·-
95%level
level
level
level
. _ - - - . -
10
91.17
8.83
LOS
9 1.17
8.83
1.34
91.17
71.31
8.83' 28.69
1.60
0.82
91.17
8.83
1.05
81.11
15.13
0.66
84.87
15.13
93.98
6.02
.83.58
15.13
78.61
17.15
0.40
88.56
10.35
0.51
_.
94.09
5.70
0.61
-
78.61
17.15
0.39
8 2.71
1 7.29
2.71
94.58 178.26
5.43
21.74
3.23
1.24
---'--
,.
0.5
20
50 .
r.~n
I
10
17.29
.~
2.11
- - _. . . . .
--------~.
0 _ _' -
••
-80%
0 •• _
28~69
1.21
91.17
8,83
.1.60
84.87
15.13
84.87
15.13
84.87
15.13
84.89
15.13
88.56
10.35
0.49
94.21
5.70
0.59
78.61
17.15
0.40
78.26
21.74
1.60
78.26
21.71.
1.90
82.71
17.29
2.82
~
-
.
_
o
71.31
_
91.17
8.83
1.25
__ .__ ~_~.Q_~L~ __.__~_2}__Q.:!.?
-.;;:-...=.:~-
y 74.7~·~;--86~2
94~23
level
•
g_L~?._._oJ:~oQ..!.
!
90%
level
91.17
8.83
1.98
~
I
93.98
6.02
0 • 99__
1. O~
89.16 94.27 i
10.35
5.70 Jl
0.52
0.62
------_.
82.71
17.29
3.73
I
94.57
5.43
4.60.
I
86.84 86.84 94.23 1
13.16 13.16
5.77
1.46
1.90
2.30.
I
82.25 89.85 94.05
l
16.11 10.13
5.95 I
0.79
1.01
1~
- - - - -..:;==.-"-";=----------=
------- ==:.:.70==. '=--";"--:;:=--";:=-="::::-';;"-==---=-"=..:.---=-=-===-===_==1
i
I
1.0
~
184.79 86.84
20! 13.16 13.60
5.77
25.19 13.18
, 1.33
1.71.
2.03
1.14
1.46
to----j----7-7-.-9-4--89-.-1-0 ~-'-~-;--80.6-;: 89.10
50
16.11 10.13
5.95
16.11 10.13
0.76
0.98
1.17
0.73
0.94
Note:
i
10
76.21
23.79
3.93
91.4 6
8.5 4
5.0 4
91.46
8.54
6.01
29.03
70.97
1.46
54.44
45.56
1.88
54.1.4
45.56
2.24
76.21
23.79
6.06
91.46
8.54
8.04
91.46
8.54
9.92
20
80.88
19.12
3.18
90.7 8
9.2 2
l~ .0 8
90.78
9.22
4.86
79.45
20.55
2.10
79.45
20.55
2.69
0-
89.35
10.65
3.21
80.88
19.12
4.01
90.78
9.22
5.22
9C.78
9.22
6.32
50
84.56
12.59
1. 70
92.2 7
7.5 6
2.1 8
7.56
2.60
19.64
1.55
87.36
12.59
1.99
92.44
7.56
2.37
80.36
19.64
1. 79
87.41
12.59
2.31
92.44
7.56
2.77
I
l
I
13.18
1.74
94.04
5.95
1.12
-
-----------
n'4j17.51
-
_.. ----
._-~--
)
The first entry in each cell is the actual probability uf the interval
covering the true value of ¢ and the second is the probability that it
will fall completely below ¢. The third entry is the expected length
of the interval.
Table 11
,..
Bias in ~ "and ~ with
-------_.-.......,....P --
va-1u~"--I-----~~~~' in ~-~-=-~.of
A
~o order n- 1 ---r-TO order n- 2
---------1-----
- --.......-----
n
=5
"-~--_._--_. "-----~----i
Bias in ~
Bias in A as i
P
p
to orqer n- 2
a
percenta~e
of bias in A.
--~
..
_~--
--
-----
0.5
-0.0239
-0.0223
-0.0020
9.1
1.0
-0.0345
-0.0336
-0.0012
3.5
1.5
-0.0374
-0.0313
-0.0001
0.2
2.0
...0.0359
-0.0365
0.0007
2.0
2.5
-0.0322
...0.0332
0.0012
3.6
3.0
-0.0276
-0.028R
0.0014
4.9
3.5
-0.0229
-0.0241
0.0015
6.2
4.0
-0.0184
-0.0196
0.0015
-------
. ...
I
__ 7'5._J
.
..
Table 12
--Second Order Moments of A and Ap for Selected Values of A with n • 5
~
Value
of A
II
I
I
I
-
~
V (A) and E (A-A)2 to various orders
se
se
V (~)=E(~ -A)2
se p
p
A
Vse(A) and
E (A-A)2 to n
se
-1
-2
A
Vse.(A) to n
to order n
E (A-A)2 to n -2
se
~
A
Vse (\)
"
Vse(A)
-2
to n
-2
E
~e
(A -A)2
p
1
.
- - - A -
E (A-A)2
se
-2
to n
nLE
se
(A-A)2
0.1
0.0387
0.0343
0.0343
0.0388
1.132
1.131
1.128
0.3
0.1091
0.1001
0.1004
0.1098
1.097
1.094
1.087
0.5
0.1716
0.1617
0.1623
0.1731
1.034
1.032
1.057
1.0
0.3024
0.2976
0.2988
0.3054
1.013
1.011
1.5
0.4095
0.4118
0.4132
0.4130
1.001
1.000
0.991
2.0
0.5035
0.5117
0.5130
0.5067
0.995
0.994
0.981
2.5
0.5911
0.6034
0.6044
0.5937
0.992
0.991
0.978
3.0
0.6765
0.6909
0.6917
0.6784
0.9:H
0.990
0.978
3.5
0.7619
0.7772
0.7777
0.7632
0.991
0.991
0.980
4.0
0.8487
0.8638
0.8641
0.8495
0.992
0.991
0.982
.
,
II
1.012
i
e
-,
•
e
<:
,
e
Figure 1
Bias in the three estimates
A
A
~, ~p
and
~*
Bias
0.8
Series to
order n.- l for
¢
0.6
0.4
0.2
10
20
30
40
50
0.0 t----t.:.:---~;::===.....;.;...-_=~::===:---+
-0.2
-0.4
-0.6
-0.8
-1.0
Sample
Size
•
Figure 2
The variance of $, $p and ¢*
•
Variance
•
2.0
1.6
.
1.2
0.8
A
</>
</>*
0.4
10
Sample
Size
Figure 3
Mean squared error of the three estimates
Mean
squared I
error
2.4
•
2.0·
1.6
1.2
.
0.8
0.4
10
20
30
40
5
Sample
Size