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POOLING CORRELATED OBSERVATIONS: BAYESIAN AND
PRELIMINARY TEST OF SIGNIFICANCE APPROACHES
by
D. R. Brogan
University of North Carolina at Chapel Hill
J. Sedransk
University of Wisconsin
Institute of Statistics Mimeo Series No. 769
September 1971
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Abstl"act
It is desired to estimate
~y
given (1) a l"andom sample of size n
from the bivariate normal distribution \'lith mean vector
coval"iance matrix
~,
!!..::(~y,llx)
and
(2) an independent random 'sample of size n y from
the univariate (marginal) normal distl"ibution \'lith mean lly and variance
2
0y' and (3) an independent random sample of size n X fl"om the univariate
2
(marginal) normal distl"ibution with mean llX and variance oX'
There is
assumed to be strong prior evidence that llX is in the neighborhood of
~y'
Thus, a preliminary test of significance approach (with H :
o
may be used to estimate l1 y '
llX::~Y)
Alternatively, a Bayesian approach to thi.s
estimation problem may be employed.
In this paper, comparable estimators of l1 y are obtained using each
of the methods (Bayesian and preliminary test of significance).
From
a set of numerical examples, these estimators are compared using mean
square error as the main criterion.
F~r
the Bayesian approach, expressions
are given for the choices of the sample sizes (n,nX,n y ) which minimize
the posterior variance of l1 y (subject to a given linear cost function
and budget).
Several applications are suggested in the Introduction.
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1.
Introduction
In many investigations, information about an unknown parameter is
available from two or more data sources.
Thus we may have measurements
Yl , ••• 'Ym for the variable of main interest Y, and Xl"" ,X for another
n
(related) variable, X.
E(Y)
= ll y '
Assuming that the investigator wishes to estimate
he may have a strong prior opinion that ll
. borhood of ll y '
X
is in the neigh-
If so, it is often preferable to use both (Xl"" ,Xn )
and (Yl, .. .,Ym) when estimating ll y '
Letting (Yl, ... ,Y ) and ( \ , ... ,X )
m
n
denote independent random samples from F(y) and G(x) respectively, a
widely used approach is to perfor.m a preliminary test of significance
of the null hypothesis ll
X
= lly'
If this hypothesis is accepted, all
-of the m+n observations are used to estimate lly whereas only (Yl, ••• ,Y )
m
is used to estimate lly if the hypothesis is rejected.
For example, in
a recent paper, Han and Bancroft (1968) have investigated the properties
of such a procedure when the
t\o10
independent (random) samples are from
two normal distributions with common (but unkno\o1n) variance.
In some applications, not all of the measurements (Yl, ••• ,Y m,Xl'" .,Xn )
"Till be independent.
For example, in a morbidity survey, a questionnaire
may be completed by all respondents \olhile each of a very small subs ample
of the respondents is subjected to a thorough examination by a physician.
Denoting by Y and X the responses to the same question as elicited,
respectively, by the physician and from the questionnaire, both Y
an~'l.
X "lill be determined for each person in the subsample but only X will
be evaluated for the other respondents.
That is, He may have a bivariate
,
random sampJ.e of size
10,
and an additionaJ., independent, random
san~le
of
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~
2
size n-m "1here only X is measured.
(The per unit samplinr; cost Houlel
usually be smaller for X than for Y.)
that
~X
is in the neighborhood of
of the null hypothesis that
~y
~Y'
= ~X
If one has a strong prior opinion
a preliminary test of significance
may be performed.
y
m
If the hypothesis
m
r
x L
is accepted, a Heighted average of
=
y./m,
= x./m and
n
m i=l ~
m i=l ~
i
=
x./(n-m) is used to estimate ~y; whereas if the hypothesis
n-m i=m+l ~
is rejected, and the correlation between Y and X is assumed to be
L
reasonably large and positive, a regression estimator is employed.
Assuming that the bivariate sample is a random sample from a bivariate
normal distribution, this preliminary test of significance approach
has been studied extensively by Brogan and Sedransk (1971).
Note,
hOHever, that in the latter paper, the sampling procedure considered
is substantially more complex than the one indicated above.
In those situations \'There preliminary tests of significance have
been extensively used by experimenters and their properties studied by
statisticians, a Bayesian approach appears to be a natural alternative.
"'Here ,one may incorporate into the analysis in a formal Hay the (strong)
prior opinion that, fOl' example, PX is in the neighborhood of lly' and
also may easily determine the consequences of one's prior assumptions.
Given two independent random samples, each of size n, from two normal
distributions with common (knovm) variance
0
2
,
f·10steller 0.9 1+8) employed
a Bayesian analysis to obtain an estimator fOl' lly'
b
= lly
on lly'
- llX - N(O,a 2 0
2
),
He assumed
and placed a localJ.y uniform prior distribution
Then t he obtClined the "maximum likcl.illood" estimator of lly' and
compared this estimator (using
exp~~~
mean squar'c error as the cri tCl.'ion)
"lith an estimator defined by the custorai.ll'y preliminary test of significance
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procedure.
In their Chapter 2313, Pratt, Raiffa and Sohlaifcr (1965)
discuss, in a more formal Hay, a problem similar to that considered
by NosteJ.ler
(191~8).
They assume two independent random samples from
tHO normal distributions:
variances are (a~,o~).
prior distribution to
the sample sizes are (ny,n
x)
and the known
They assign an ax:-bitrary bivariate normal
(~y,llX)
and perform posterior and preposterior
analysis, but do not compare their results Hith those obtained by using
a preliminary test procedure.
rTe extend the results cited above by considering a more general
sample design.
Thus, He assume (1) a random sample of size n from
the bivariate normal distribution with mean vector
_
covax'iance matrix
ja~
>::
1
1:.::
(~y ,~x),
and
poya
a
,pay X
02
X
XJ ,(2)
an independent random sample
of size n y from the (univariate) normal distribution with mean
variance o~, and (3) an independent random sample of size n
X
~y
and
from the
(univariate) normal distribution with mean ~X and variance a~.
As
indicated by the example presented above, this type of sample design
is common to many sample surveys.
To simplify our results and to make
them parallel ".lith those given by Brogan and Sedransk (1971), He assume
that the components of
~
are knO"m.
normal prior distribution to
(~y,lJX)
He assign an arbitrary bivariate
and carry out the posterior analysis.
- lJtilhing the sample design defined by (1), (2), (3) above, our main
objective is to compare two types of estimator of lJ y :
(1)
the posterior
expected value of Py' and (2) the estimator obtained by first applying
a preliminary test of
significanceo~
tho null hypothesis lly :: \-lX' and
then using an appropl'iatc ostima tor depending upon the outcome of the
test.
For our comparisons, He consider (Sections 2.2, 2.3)
tYlO
bivill'iatc
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normal prior distributions Vlhich conform to the assumptions underlying
the use of the preliminary test of significance approach.
If the preliminary test procedure is used Vlhen the sample design
consists of a random sample of n observations from the bivariate normal
distribution t and
t\-lO
independent random samples from thn tHO univariate
nonnal distributions t it must be assumed that the components of
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are
knoHn if usable analytical results about the properties of the estimator
are to be obtained.
Thus t for purposes of comparison t the components
of L are assumed to be knmm in the Bayesian analysis.
The treatment
of unknown L Hi thin the Bayesian framm'1ork is discussed very briefly
in Section 5.
...
Assuming a linear cost function and a given budget, one may
determine those values of n, n
variance of
~y.
X
and n y \-Ihich minimi2e the posterior
The procedure to obtain this optimal allocation is
given in Section 4.
It may be noted that if one uses the preliminary
test approach, obtaining the optimal sample sizes is a much more
formidable task [see Brogan and
Sedrans}~
(1971)J.
Finally, in many sample surveys repeated on two or more occasions,
one uses a "panel" of indivi'duals Hho are replaced over time.
Thus,
one may have a sample of (n + n ) individuals for the first time period
X
of 'whom n are reta.i..ncd for the second time period.
Then,. excluding these
(n + n ) individuals, an additional n y individuals are selected from the
X
population to enteX' the panel for the second time period.
(Thus, the
sample for the second time period consists of n + n y individuals)
Problems such as determining aTl appropriClte os titnator for 11 y (the
population mean of Y for the second time period), and
dete~nininE
how
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many
individuals are to be retai.ned in the sample (Le.,
_
....-- the optimal
..
_.
value of n) are discussed (Hit;hout any distribution assumptions) by
Cochran (1963; Sections 12.9, 12.10).
However, the sample design con-
sidered in our paper is identical to that described above for sampling
. on two occasions.
AlSO, the bivariate normal prior distribution for
(Vy,V ) can reflect the prior opinion that Vy and Vx are closely
x
related. Thus, if the assumption that (y,X) folloHs a bivariate normal
distribution is a reasonable one, the Bayesian anaJssis to be presented
in Section 2 yields an appropriate estimator for Vy ' and the results
in Section
II
give a procedure for determining the optimal value Of n.
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2.
Posterior Analysis
2.1
The model and basic analysis
Consider a random sample of size n from the bivariate normal
distribution with
EC
vc
~y
)
=
C:)= ~.
=
C'p:ya
llX
~y
llX
)
and
~yax)
er 2
X
:: _l
X
and two independen,!. random samples (of sizes ny and n ) from the two
X
univariate normal distributions defined by the marginals of the bivariate
The random variables (Y1 t • • • t Y11 t Yn+ 1 t ••• t y
11+n.'
y
are to be used to determine an estimator for
X lt • • • tXt
n Xn+ 1 t ••• t Xn+n )
X
lly; here (Y.tX.) for i=l, ••• ,n denotes the bivariate sample. Assuming
normal distribution.
~
~
E kn~wn t i t can be shovm that (Qy,QX) are joint sufficient statistics
_
n
_
n+n y
_
where Yn ::
y./n t Y =
y./11 yt x
n
i=l ~
ny i=n+l ~
l
r
n
= L x./n,
i=l ~
_
and x
n+n
= L X x./n X
n X i;n+l ~
Qx is obtained from Qy in (2.3) by interchanging X and Y.
It is easily
shown that, given (lly,llX)t (Qy,Qx) has a bivariate normal distribution
with
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E
(Y
(Y
lly
Q
llX
X
V
~Y)
) (
:::
ll'
Y
Q
llX
X
where
)
:::
and
ll'
X
(S,
8:
S~)
y
SXY ::: RxySXS y ,
(2.5)
S 2
X
~\
::: [n y +n/(I-p2)J, and
One obtains (S~,MX) from (S~,My) by inter~hanging
X and Y.
First, He consider as the prior distribution for (lly,Jl ) the
X
hivariate normal distribuU.on with
E
CY) (::)
V
C:) C'
:::
llX
=
Y:Xoy
,
and
(2.6)
•
(2.7)
yo.x"Y)
0. 2
X
Those special cases of the prior distribution which seem to be the most
appropriate for comparisons Hith the preliminary test approach Hill be
considered in Sections 2.2 and 2.3.
Using (2 .Il) through (2.7), it can be shovln that the posterior
distribution of (lly,llX) is bivariate
nOl~mal
with
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+ -----
+ [Qy+RxySy(bX-QX)/SX]
S~a~(1_y2)(1_R2
y X
Xy
)
+ [by+yay(QX-bX)/a X]
- ..
S~a~(1-y2)(1-R~)
(2.8)
+
(2.11)
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Qn~ may obtain E(vxIQxtQy ) and
V (llXIQXtQy) by interchanging y and X
in (2.8) and (2.10).
In Sections 2.2 and 2.3 two special cases of the bi vaPiate normal
prior distribution are considered.
Then t in Section 2.4 t there is a
brief discussion of the form of t(vyIQytQ
x)
as given by (2.8)t (2.9)
and their counterparts in Sections 2.2 and 2.3.
2.2
Bayesian "normal" analysis
In Section It it was asserted that the use of the preliminary
test of significance approach implies a strong prior belief that V is.
x
in the neighborhood of yy.
To reflect this prior belief t i t may be more
convenient to assess a prior distribution on (6=V y-V tll y ) rather than
x
on (Vy,V ),
X
Here t it is assumed that,
~pri0X:ft6
and Py are (independently)
normally distributed with B(6)=Ot V(6)=a 2 and E(Py)=b y ' V(lJy)=Ct~.
It
is easily seen that this specification is equivalent to assuming that
(l1 y ,llX) folloHs the bivariate normal distribution given by <2.6) and
(2.7) with bX=b yt Y=Cty/a
x and Ct~=a2+Ct~. Theri t the posterior expected
-value and variance of 11 y are given by:
= n-1[by {[a y2 a 2 J-l
(2.12)
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(2.13)
.. where
2.3
Bayesian "precise measurement" analysis
An alternative prior distribution is obtained by taking a
locally uniform prior distribution for Py , and by assuming, as in Section
2.2, that fl_H(O~a2).
This prior distribution corresponds to the notions
that both Y and X are estimating the same
mean is "unknovlD"
~
priori.
mean~
but the value of this
This might be the reasoning of one who pro-
poses to use the preliminary test approach to esttmate }ly.
expected value and variance of Py may be obtained from
and (2.H) by letting a~.)o
2.4
+00.
The posterior
(2.12)~
(2.13)
Then~
Discussion
Each of the estimators of}.1y [see (2.8), (2.9,) (2.12) and
(2.15)J combine the prior and sample information in a sensible way.
For
instance, the estlmator given in (2.8) is a linear combination of the
pr'ior expected value of lly, by; an unbiased estimator (and component of
the joint sufficient statistics) of
on OX; and a linear'
l~egression
}.1y~
Qy ; a linear regression of Qy
of by' on b "
X
,
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From (2.B) and (2.9)
it is clear that ~s the prior variance of Vy~ a~~
is increased, the Height on by' the prior expected value, is decreased.
Similarly, as Sy 2
= Var( Qy IV X ' 11y)
is augmented, the weight on Qy is
The weight on the regression of Oy oJ.1 Q is decreased if s~
X
reduced.
or Ct~ or (1_y2) or (l-R~) is increased.
regression of by on b
increased.
X
Similarly, the Height on the
is reduced if s~ or a~ or (l_y 2) or (l-RX~) is
Each of these conditions appears to be a natural consequence
of the information contained in the prior distribution and in the likelihood function.
Ruhl (1967) discusses in detail the posterior expected values
obtained by applying Bayesian "normal" analysis (Section 2.2), and
Bayesian l'precise measurement" analysis (Section 2.3).
expected values are easier to interpret than (2.8).
cases are considered:
and (2)
(1)
These posterior
Also, some special
p=O, n=O (Le., two independent samples),
the limiting cases of E(V y IQX,Qy) as a 2 -)- +00 and as a 2 -"0.
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3.
Comparisons of the preliminary test and Bayesian approaches
3.1
Introduction
-To compare the preliminary test and Bayesian approaches, we
have used the sample design specified in Section 2.1 with ny=O.
Consider
lly;lPX using 7,=(1 n -x n+n ) as
X
the test statistic. Here, x
=
(nx+nXx
)/(n+n
)'
If
is
accepted,
n+n
n
n
X
3
X
X
is used where c. :: h.1
h., and
the estimator 0YO = c X +c y +c x
1 n
2 n 3 nX
~
~ j::1 J
"0
I
h
h
2
::
2
n[(oy-2) - (poX-1 0Y -1 )J/(l-p),
2
3
-1
n[(oX-2)
1
h
-1
::
(pax
0y
)J/(J.-p),
= (nxl°~)'
( 3.1)
If H is rejected, the regression estimator 0YR=Y + [poy·
n
O
employed.
(xn+n -xn )/oxJ
is
X
The estimator defined by this procedure will be denoted by 0Y'
[For further details about this preliminary test procedure see Brogan
and Sedransk (1971).]
-Pr(Reject HO)=l,
to the
Now, if the
Oy=OYR'
prelimina~J
\1e consider both
test is made so that
Oy and 0YR as alternatives
posterior expected values given by formulas (2.12) and (2.15).
For ease of reference, denote by 0YBN the expression for E(llyIQy,Q x) in
(2.12), and by 0YBP the expression in (2.15).
- The quantity used to compare
erl:'or.
Oy' 0YR,OYBN and 0YBP
~s mean square
Roberts (1966) discusses some of the problems inherent in an
investigation such as this one, and suggests cri teT'ia Hhich the quantity
used to compare the estililOltors should meet.
Assuming squared error
loss, Roberts' criteria are satisfied if the mean squar.e error of each
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13
estimator is determined for several values of t, and lly'
"true" (but unknmll1) value of lly-llX'
Listed beloH are the expressions
for the bias and mean square error (or variance) for each of the estimators.
The ones for fly and 0YR are given by Brogan and Sedransk (1971):
t 0:-0
Bias (Oy)
= J [-ooZ(c
-E; -0
1
+c ) - (II 2 toz-1)]epCt)dt t
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3
0:
\
(3.5)
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Beret t, is the
ta
J ¢(t)dt = I-at
-~a
the c. and h. ape def5.necl in (3.1),
~
J.
and
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It is a straightforward algebraic task to find the bias and mean
square error for each of 0YBN and 0YBI':
Applying (2.4) and (2.5) to
the' definitions of the estimators [formulas (2.12) and (2.15)J, one
obtains
where D is defined in (2.14).
Similarly,
(3.8)
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where D* ~ a-2{[s~2J + [S~2J - [2R Xy /S Sy J} + [SXSyJ-2.
X
In Ruhl (1967) there is some intel~pretation of these biases and variances.
Two h~teresting limiting cases (a 2
3.2
-)0
+00, a2.~ 0) are also considered.
Description of the numerical examples and tables
It is apparent: from formula (3.5>" that an analyticul comparison
of the mean square errors of
Oy, 0YR'
flYDN and 0YBP is not feasible.
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Thus, we have considered the thirteen numerical examples having values
of (n,llX,(J~,(J~,p) given by (9,18,6,12,.10), (9,18,6,12,.33), (9,18,6,12,.60),
(50,200,6,l2,.10), (50,200,6,12,.60), (9,36,6,6,.10), (9,36,6,6,.60),
(25,50,6,6,.10), (25,50,6,6,.60), (9,18,6,24,.10), (9,18,6,24,.60),
(25,50,6,24,.10), and (25,50,6~2lf,.60). It is assumed throughout, Hithout loss of generality, that lly=O.
For each example He took 0:=0.50,0.25,
0.10 for the preliminary test approach, and,also, 6=0,1,2,3.
[Here, again, 1'1
refers to the "true" value of lly-llX; recall that Hi th the two Bayesian
°
approaches I'1-N( ,a 2 ) . ]
For the Bayesian methods we considered all eighty
combinations of a 2 , by and o:~ Hhere a 2 =0.25,1.00,2.00,4.00;
b y =0,±1,±2; C/,~=0.25,1.00,2.00,4.00. Thus, for the Bayesian "precise
measurement" approach there are four values of a 2 and four values of 1'1
- for each example.
Similarly, for the Bayesian "normal" approach there are 320
combinations of 6, a 2
,
by and C/,~.
It should be noted that only non-
negative values of 1'1 need to be considered since for 1'1>0, Bias WyBP (-I'1)] =
-Bias [OYBP(I'1)]; M.S.E.[PYBP(-6)] = M.S.E.[OYBP(6)]; Bias [OYBN(-6,b y )] =
-Bias[OYBN(I'1,-b y )]; and M.S.E.[OYBN(-6,by )] = M.S.E.[OYBN(6,-b y )].
for example, Bias[OYBN(6,-by )] and
Here,
l'1.S.E'[~YBN(I'1,-by)] denote, respectively, the
•
~
b ~as
and mean square error of llYBN
evaluated at (6,-b y ).
Finally, Brogan
and Sedransk (1971) shoH that·M.S.E.[Oy(6)] = M.S.E.[D y (-6)] and Bias[P y (-6)] -Bias[p y (6)].
•
For each example and each combination of 0:, 6, a 2 , by and
bias and mean square error of each of
culated.
Oy'
1O:y'
the
0YR' 0YJ3N and 0YBP were cal··
Some of these results are presented in Tables land- 2, and
are discussed in Section 3.3.
In Table 1, n.=25, n =50, (J~~6, (J~::24,
X
P=.60 while in Table 2, n=50, n =200, 0;=6, 0~=12, p=.60.
X
In each
table the mean square Cl'ror'S of 0YhP are g.i. ven in the first l'O'.:.
Then,
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16
corresponding to the twenty different values of (by,a~), the mean square
er~ors
of
PYBN
are presented in the succeeding twenty rows.
the mean square errors of
PYR
are given.
Py
Finally,
(for each of a=O.lO, 0.25 and 0.50) and
The column headings give the values of ~, and, where
appropriate, those of a 2 •
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17
Table 1.
Mean square errors of ~j.lYBP' f'lPYBll' ~
Py an<l ~
f.'YR \lllen
'
n=25,n =50, a~=6, a~=24, p=0.60
X
!:J.=O
I
----I
-'---r
a 2 =0.25
a 2 =;L.00
a 2 =2.00
0.16314
0.17236
0.17637
0.17906
0.05805
0.11936
0.13869
0.15017
,0.22083
.0.22083
0.14028
0.14028
0.1 1{477
0.14477
0.15182
0.15182
0.70916
0.70916
0.20303
0.05916
0.124'18
0.l'I551
0.15808
0.23065
0.23065
0.14703
0.15210
0.15210
0.15987
0.15987
0.7 1{512
0.74512
0.21468
0.05985
0.12685
0.1 1+856
0.16157
O. 231~14
0.23 IH4
0.14993
0.1'+993
0.15532
0.15532
0.0603 11
0.12845
0.15062
0.16392
,0.23632
0.2363?
0.15187
0.15187
"O~20303
0.21'168
'0.16300
0.16300
0.15675
0.15675
0.17187
0.17187
0.16524
0.1652 11
M.S.E.(OYBN)
«2
by
Y
0.0
0.0
0.0
0.0
1.
... )..
1.
-1.
1.
-1.
1.
-1.
2.
-2.
2.
..2.
2.
..2.
2.
..2.
0.25
1.00
2.00
4.00
0.25
0.25
LOO
1.00
2.00
2.00
'1.00
11.00
0,.25
0.25
1.00
1.00
2.00
2.00
11.00
11.00
0.1W103
6=0
B.S.E.<Oy)
«;:0.10
«=0.25
a=0.50
0.1676
0.17'+0
0.1791
B.S.E. <V yR )
'0.182'1
0.163 Lfl
0.163lJl
0.75700
0.75700
0.21920
0.21920
0.17560
0.17560
0.16893
0.16893
o .1571~8
0.15748
0.16579
0.16579
0.76 L128
0.761~28
0.22212
0.22212
0.17808
0.17808
0.17139
0.17139
\~
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20
Table 1.
6=3
t
_ _ _a
I
a =2.00
a 2 =4.00
0.18896
0.18268
0.06 1112
0.13590
0.15916
0.17310
0.18384
0.29297
0.l3007
0.18790
0.06156
0.13105
0.15366
0.16723
0.20826
0.26683
0.13888
0.17005
0.15139
0.16966
0... 16412
0.17407
0.70693
0.82408
0.19354
0.25589
0.16284
0.19940
0.16475
0.18465
2
0~14899
0.18286
0.• 16573
0.18415
0.65214
0.870 111
0.17042
0.28607
·0.1523 1{
0.22007
o.16?Oll
0.19887
Continued
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22
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Continued
Table 2.
1
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2
----,
6.=1
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a 2 =2.00
a 2 =4.00
a 2 =0.25
0.08156
0.08245
0.08365
0.20829
0.04637
0.06973
0.07530
0.07834
0.08479
0.12898
0.06711
0.08372
0.07235
0.08132
0.07640
0.08107
0.24425
0.33262
0.07587
0.10908
0.072'17
0.090 /n
0.07527
0.08460
0.0 11635
0.07024
0.07598
0.07912
0.09728
0.12061
0.07175
0.08059
0.07519
0.07997
0.07829
0.08078
0.27339
0.32006
0.08511
0.10279
0.07761
0.08717
0.07830
0.08327
0.04675
0.07113
0.07701
0.08023
0.121172
0.18089
0.19387
0.20089
0.0 1129 11
0.30883
0.13732
0.23373
0.16928
0.22094
0.18782
0.21'159
0.06350
0.59529
2
a =0.25
a =1.00
0~10013
0.05995
0.08695
0.09319
0.09656
0.04 1165
0.17759
0.06749
0.11569
0.08152
0.10735
0.09019
0.10358
0.13168
0.3975'1
0.05730
0.15371
0.07233
. ·0.12399
0.081447
0.11.;1.2 11
fj=l
0.1.327
0.1018
0.0899
O. 0851~
o .101~ 1~6
0.116 116
0.07 1191
0.07947
O.077 l fl
0.07988
0.08001
0.08130
0.28958
0.31358
0.09080
0.09993
0.08110
0.08604
0.08065
0.08322
0~10303
0.29585
0.11~717
0.25050
0.17541
0.2289 11
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Table 2..
6=3
--'-~r-
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a 2 =2.00
a 2 : lt.00
0.09019
0.08567
0.05070
0.07683
0.08311
0.0865li
0.07830
0.1li830
0.06950
0.09602
0.07754
0.09188
0.08322
0.09069
0.23108
0.37108
0.07403
0.12706
0.07518
0.10386
0.0807't
0.09567
0.04788
0.07285
0.07887
0.08217
0.09359
0.12959
0.07206
0.08576
0.07680
0.08 1122
0.08066
0.08 1152
0.26671
0.33872
0.08339
__:0.11078
0.07802
0.0928 11
0.08001
0.08773
Continued
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25
3.3. Conclusions from the numerical examples
Our main objective is to compare the mean square errors of
and 0YR with those of 0YBN and 0YBP'
to compare 0YBN with 0YBP first.
Oy
HOI'lever, it is more convenient
If a~ is at leas~ moderately large,
the coml)arison of H.S.E.(OYBl-I) with I'l.S.E.(OYBP) Hould appear, as a
rough approximation, to depend on the comparison of
( 3.10)
with
BP
2
) =
= In*[Bias(O YBP )JI/(i-RXY
.
[See formulas (3.6) through (3.9) and (2.14).J
xy
}
SxSy
R
(3.11)
•
Assuming that {(S~2)-RxyS~lS~1}>O
(Hhich is true for all of the thirteen numerical examples), it appears
from (3.10) and (3.11) that the comparison Hill depend on (1) the mag-
by and b. have the Same sign.
To investigate, consider the numerical
example (n=25, n =5o, a~=6, a~=24, p=.~O) whose results arc given in
X
Table 1.
Then the conditions Hhen H.S.E.<!\Bl~) > ~1.S.E.(f.i'YBP) can be
8ho.111 to be, approx.imately, given by
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26
c.
Iby IICYy ~ /2 ; l.:s. Iby I/a y
6 and by have oppos-he signs:
° <lbyl/O:y
and 1 < 161/a < 3;
< 12
< 12 and 161/a ~ 3.
It is apparent that among formulas (3.2) through (3.9), only (3,6)
contains the quantity by'
Thus, instead of taking lly=O fol" OUl" numel"ical
examples, an equivalent approach would be to take
than b y =0,±1,±2.
by~
If He use by-11 y rather' than
by-Py=0~±l,±2
rather'
the conditions listed
in A, B, and C above al"e changed by r'cplaeing Ibyl/ay ,·Tith Iby-llyl/a y '
Then, since these conditions include both 6 and lly'- useful summar'ies
may be obtained by finding the pr'ior pr'obabilities of the events listed
under' A, Band C.
Using the (joint) pr'ior' distribution on (6,lly)
descr'ibed in Section 2.2, P(A)=O,0455, P(B)=O,02l7 and p(C)=O.1054.
Here, peA) denotes the pr'ior probability that Illy-by{tay ~ 2; and PCB)
denotes the prior probability that 0<161/a<2 and Illy-byl/ay ~ 2, or
2~161/a~4 and Illy-byl/ay > 4 where 6 and by are assumed to have the ~
sign.
Thus, the (pl"ior) probability that M.S.E.(PyBN»M.S.E.(OYBP) is
relatively small.
Similarly, for the example (n=50, n X=200, a~=6, a~=12, p=.60) in
Table 2, the conditions \o1hen r-l.s.E.CoyBN»r-'l.s.E,(i)YBP) arc, approximately:
Ibyl/cty ~
1:2.
A.
b=O:
B.
6 and by have the same sign:
2.:s.161/a<lj
C.
and IhyI/Cty':'l+.
6 and by have opposite signs:
1': Iby II ct y
o
0<161/a<2 and Ibyl/ay~2;
< /2 an d
< Ibyl/cty <
/2
2 -~~.:s.
Ibyl/cty':'
Ib II a
1:2;
< 2;
and lAlla ~ 2.
Proceeding as \-lith Example 1, P(A)=O-,1585, ·PCB)::O.0217 and pCC):.:0.1327,
Thus, aga in, the prior' prob2bili ty that 1·1. S, E, (flYBN) >;-l. S, E, (OYDP) is
relatively small.
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27
Bi examining the results for the other eleven examples, it is
'. apparent that for each example the approximate conditions when
M.S.E.(P yBN ) > M.S.E.(OYBP) can be expressed in the same form as given
above for the tHO "demonstration" examples.
In gc;,eral, as expected,
0YBN tends to have a smaller mean square. error than 0YBP unless Ibyl/ay
(i.e., Iby-~yl/ay) is large.
This is most clearly evident for the
important special case, 6=0.
To compare the mean square errors ~f Oy and 0YR
\-li
th those of
0YBN' \-1e try to identify the conditions ",hen n.s.E.(OYBN) > H.S.E.(Oy)
and r'l.s.E.(OYBN) > H.S.E.(OYR)'
Those values of (6/a,hy/ay) for \-lhich
~l.S.E. <1\BN) > M. S.E. (OYBP) are the values for "'hich N.S.E. (OYBN) is
largest.
one case:
Thus, it is not too surprising that in Table I (in all but
6=1, a 2 =1, b y =-2, a~=4) if M.S.E.(OYEN) > M.S.E.(OYBP) for
some value of (6/a,by/ay)' then M.S.E.(OYBN) > M.S.E.(Oy)' or
H.S.t.(OYBN) > f1.S.E.(OYR) for that same value of (A/a,by/a y ).
Such
a pattern persists for the other examples, and the exceptions correspond
u
·t6 i:lle-"less·extreme" configurations of fA/a,byla.y)'
[For instance, a
"less extreme" configuration of (6/a,by/Cty) may be one \-lhere 6 and by
have opposite signs, but both lAlla and Ibyl/ay are relatively small;
or where 6 and by have the same sign, 161/a is large and Ibyl/a y is
_ relatively small.] It is also true in most cases that if .M.S.E.(OYBN) >
M.S.E.(Oy) for a=O.lO, 0.25, or 0.50, or M.S.E.(OYEN) > M.S.E.(OYR)'
then B.S.E.(OYBN) > l'l.S.E.<1\) for all values of a, and r'l.s.E.(OYBU) >
I1.S.E.(OYR)'
Th.e exceptions are of the cxpected type:
"less extrcine"
configurations of Ul/a,Dy/ct. y );· and sin?ll values of a Hhcn !.I is small,
or larger values of n \-lhen 6 is larger.
(Recall that 0YR can be defined
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28
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.,0:
For instance <in Table 1), "lith /),=0, a 2 =1, b y =2,
a.~::2, M.S.J::.<OYBN) > M.S.E.<Oy) only for 0',::0.10; Hhile "lith /),=1, a 2 =lf,
b y =-2, 0:~=2, M.S.E'<OYBN) > I1.S.E.<OYR)' only.
Of course, there are cases Hhere I1.S.E.<OYBN) < M.S.E.<OYBP)' and
M.S.E.<OYBN) > I1.S.E.<1\) or M.S.E.<OYBN)·> M.S.E.<OYR)'
As expected,
in most of these cases, the configurations of <f:../a,by/O: y ) are of the
same ~ ~ 'S!pe as those for Hhich M.S.E.<OYBN) > l4.S.E.<OYBP)' but
. are somevlhat "less extreme."
(see, for e.xample, Table 1:
HOiolever, in a relatively feH situations
6.=2, a 2 ;:0.25, by=l, ~::I~),
/),
and by may
have the sar~ sign, 16.I/a may be very large, Ibyi/ay very small, but
M.S.E.<OYBN) > M.S.E.<Oy)' ",hile <as expected from our previous comparisons) H.S.E.<OYBl~) < fvl.S.E.<OYBP).
From the preceding discussion,
and by further investigation of the numerical examples, it appears that
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"I
as fly "lith 0:=1.)
-the conditions \olhen H.S.E.<OYBN) > B.S.t.<Oy) can be summarized (in a
very general Hay) as folloHS:
A.
/)"b y have the same sign and 1</),/a)
-
(hy/el y ) 1 is large.
For
a given value of I/),I/a, this usually means that lbyl/a y is lal'ge.
B.
/)"by have opposite signs, and, for a given value of I/),I/a,
Ibyl/o:y is "large."
As the given value of I/)I/a is increased,
the value of Ibyl/ay for "Ihich M.S.E.<OYBN) > !'l.S.E.<Oy)'
decreases.
Thus, when 16.I/a is quite large, M.S~E.<OYBN) >
M.S.E.<Oy) for all values of Ibyl/cv'Y'
C.
6.::0.
I1.S.E.(OYBN) > H.S.E.<Oy) if
I byl /a y
is SUfficiently
large •.
Hhen comparinG
6.>0 and /),::0.
•
Oy
and 0YR Hith l\Bp s tHO cases must be distinGuished:
From the example~, it is clear that (for /\>0) if II\\/a is
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29
sufficiently large, H.S.E.<(tYBP) > H.S.E.<f\' for at lea~t one of
- a::O.10, 0.25 and 0.50, or t-1.S.r..<OYBP) > H.S.E.<OYR)'
If 6=0,
M.S.E.(OYBP) > M.S.E.<Oy) for at least one value of a Hhen a 2 is
sUfficiently large.
a2
HOv1ever, for
is large and a is small do
instance, in Table
vIe
1:,::0,
it is often true that only when
have M.S.E.<Oy) < t1.S.E.<OYBP).
For
1, M.S.E.<Oy) < M.S.E.<OYBP) for a=O.lO when a 2:1,
and H.S.E.(Oy) < t1.S.E.<OYBP) for a::0.25 when a2~2.
Since large values
of a 2 denote considerable prior uncertainty about the value of
would seem to "correspond toll large values of a.
I:,
they
Thus, although a
calibration of respective values of a and a 2 is beyond the scope of
this paper, it should be noted that, for a given value of a, M.S.E.<Oy)
may exceed H.S.E.<OYB ) for all of the lIcomparablell values of a 2 (e.g.,
P
-2
in Table 1, let a=O.lO lIcorrespond toll a ::0.25).
This difficulty of equating values of a 2 and a was not considered
\-lhen·we compared 0YBN with
Oy
because meaningful, overall, conclusions
about the relationships among the mean square errors can be made by
---determining the conditions under which M.S.E.<OYBN) > M.S.E.<oy) for at
least one value o~,!, or t-l.S.E.<OYBH) > 1-1.S.E.<OYR)'
Of course, such
overall comparisons "favor" the preliminary test approach.
Hith the
latter method a si!2f2:! value of a must be selected, and i t is 'dell knovTn
that a small value of a is most advantageous (in terms of small mean
square error) \'lhen IY. is small \-Ihile a large value of a is best "Ihen b.
is at least moderately large [see, for example, Ruhl (1967; Chapter 4)
Or Han and Bancroft (1968) J.
Thus, Hhen determining the coneli tions
under Hhich H.S.E.<OYBl-I) > H.S.E.<Oy.) for at leClst one value of
0.,
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30
M.S.E.<OYBN) is being compared (for a specified value of 6) with the
minimum (taken over the possible values of a) value of M.S.:c.(Oy) for
that value of 6.
HOHever, the conclusions dravm do indicate the general
relationship bet"leen 11.S.E.(OYBN) and H.S.E.(Oy) because, as noted
earlier, when 11.S.:c.(OYBN) > M.S.E.(Oy) for at least one value of a,
it is likely to be true for all values of a [especially for the more
extreme configurations of UVa,by/ay)J.
To illustrate the general statements made above, we first consider
the example (n=25, n =50, O~=G, O~=2Lf, p=.60) in Table L
X
The conditions'
.when M.S.E.(OYBN) > M.S.E.(Oy) (for at least one of 0.=0.10, 0.25, 0.50),
or ~l.S.E.(OYBN) > lvl.S.E.(f\R) can be shovlrl to be, approximately, given by
A
l
•
6 (1\;[0) and by have the same sign, or by=O:
and Ibyl/ay ~ 2; 2 ~
161/a
< 4 and Ibyllay ~ 4;
(Ibyl/ay ~ 4 or 0 ~ Ibyl/ay ~
B
1
•
0 <
161/a < 2
161/a ~ 4 and
1:2).
6 (k/-O) and by have opposite signs:
Ibyl/C'y >
12;
0 < Ibyl lay <
161/a ~ 2.
Iby Ilay ~ 12.
. c 1 • 1i=0:
and
Note that the results cited in A , B , and C
1
1
for ,instance, Hhen 6 and by have opposite signs
pattern cited earlier.
and C above, He note that the prior
a summary of the conditions in A ,B
1
probabilities are:
above conform to the general
1
1
1
P(A )=0.0217, POl )=0.098 I f, and P(C )=0.1585.
1
1
Thus,
1
the prior probubili ty thut 1\Bl~ is inferior (in terms of mean square
error) to Oy (or 0YR) is relatively smalL
Similarly, the conditions Hhen .1'l.S.I:.~OYB1) > H.S.E.(Oy)' or
H.S.E.(OYBP) > r.l.s.E.(t\H) are, approximately, given by
12
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D.
1:.'10:
It:.l/a.~ 1.5.
E.
/:'=0:
for a=O.lOt a2~1; for «=O.25 t a 2>2.
1
.
1
= 0.1336.
Note that P(D )
1
The results exhibited in A through E
1
1
obtained from most of the thirteen examples.
above are typical of those
HOHever t for several of
the examples t the pattern is slightly different.
Thus t He nc>:t consider
the example in Table 2; the results summarized beloH (for n=50 t n =200 t
X
C1~=6 t C1~=12 t p=. 60) are typical of the results for 'the remaining examples •
For example 2 (Table 2)t the conditions Hhen 1.1.S.E.(f!yml) > H.S.E.(f\)
(for at least one of a=O.lOt 0.25 t 0.50)t or 1.I.S.:c.(OYBU) > 1.1.S.:c.(OYR)
can be shown to be t approximately t given by
I:. (1:.'10) and by have the same sign t or by=O:
A •
2
o
< It:.l/a < 2
and Ibyl/ay > 2; 2 ~ II:.I la ~ 3 and ( 1by 1lay > 4 or
o
B
2
.
< Ibyllay < 0.5); II:.I/a> 3 and 0 ~
I:. (/:''10) and by have opposite signs:
1 < Ibyl/ay <
-
·~and
1:.=0:
C •
2
12 and 12< 11\1 fa <
2-t
Ibyl /Cl,y < 2.
Iby II Cl y ~ /2";
o
< Ibyl/ay <
1:2
1I:.I/a ~ 2.
Ibyl/ay ~
o.
Hhile the pattern exhibited in A is slightly different from that
2
exhibited in A
t
the pattern in A is consistent with the general pattern
1
cited earlier.
2
It may be noted that for most of the
spec~ficati.ons
having 1:.=0 and a 2 =0.25 t N.S.E.(OYBN) > H.S.:c.<oy) only for
FinallYt the prior probabilities are:
p(c )=1.
2
peA )=0.0312 t P(}3 )=0.107 11 and
2
2
Hote that peA ) • peA ) and 1'(13 ) • PCB ).
1
2
(~:::0.10.·
1
2
.
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32
Similarly, for example 2, the conditions vlhere H.S.E.<OYllP) >
H.S.E.(Oy), or ~I.S.E.(DYBP) > I1.S.E.(OYR) are, approximately, given by
D.
At-o:
lAlla>- 12.
E.
~=O:
for a=O.lO, a 2 >0.25; for a=O.25 and a=O.50, a 2 >1.
2
2
Note that P(D ) = 0.1585.
2
-
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33
. 4.
Determination of the optimal sample sizes
Consider a bivariate random sample of size
random samples of sizes
nX~O
and
ny~O
n~Os
and tHO independent
as described in Section 2.1-
Assuming the bivariate normal prior distribution on (11 s)1 ) given by
y X
(2.6) and (2.7)s we wish to find those values of n>Os nX>O and n >0
-
which minimize
-
y-
VCl1 y IO x,Oy) [see (2.10)s (2.11») subject to
where C* is the given budgets Co is the fixed overhead costs c Xy is
the cost of measuring both X and Y on a single units cY is the per
unit cost of measuring only Ys and C is the per unit cost of measuring
x
only X.
usually,
It is assumed that O<cX::cxy::cX+c y and O<cy::Cxy::cX+c y Hhile,
CX~Cy.
Clearlys the optimal sample sizes
the prior distributions
assu~cd
correspondin~
to
in Sections 2.2 and 2.3 can be easily
obtained from the results to be presented beloH.
To find the optimal values of n~Os nX~Os ny~Os we proceed in tHO
steps.
Firsts n is assumed to be fixed and the optimal values of n
X
and ny are determined (as functions of the fixed s but arbitrary, value
of n).
Then s the optimal choice for n is obtained.
Assuming
n
fixed
)s it is more convenient to consider the quantities U
Xy
and V rather than n and n y :
X
[O<n«C*-c )/c
- -
0
I:
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22222
where (XX'
Uy '
that Hx/cr~
y, HX' 11y ' cr x and cry are defined in Section 2.1.
= [S/(l-RX~)]-l.)
(Note
Similarly, define
Since n y ~ 0 and n ~ 0, 11 y ~ nB and H ~ n(3.
X
X
Therefore,
(4.5)
vn •
Using (1.1.1) through (4.6) Hith (2.10), (2.11) and the definitions
given in
~ection
2.1, it is easily seen that (given
find those values of U > U
n) we wish to
> 0 and V > Vn -> 0 such that
n -
V(lI
~y
10y' 0)X -
U/[UV-H 2) is minimized subject to the budget restriction
n
cU+cV=c
1
2
n
Thus, using (LI. 7),
f(U) = c VI[c II - c V 2
2
n
1
\otish to find that value of U Hhich mlnil:lizes
He
-
C \-12)
2
n
subject to V < V < (c -c V )/c.
n n 2 n
1
investigating df(U)/c1V, i t is easily ·~een that the optimal sa:npling
procedure is:
By
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35
(1)
Ie 2 Ic 1 1\"1n I -> (cn - c 2 vn )/c 1 ; take U = (cn -c 2 Vn )/c 1 ,
If
V
= Vn
(~,
(II) . If Ic I c IW
2
1
n
(III)
Othe~~ise;
(4.8)
ny=O).
I
< U ; take U = U (i.e., nX=o), V
n
n ----
-
take U = Ic Ic
1\"1 I,
n
21
V
=. (cn
= (cn-c 1 Un )/c 2 •
Ie'CIH I )/c •
-
n
12
2
The optimal sampling procedure as given by (4.8), (4.9) and (4.10) can
be obtained from the results given by Pratt, Raiffa and Schlaifer (1965)
on pp 45-46 of their Section 23.12.2; with n fixed, our problem is
closely related to theirs.
To find tile optimal value of n, one must consider separately each
of the three cases corresponding to (4.8), (4.9)t and (4.10).
case we find that value of n
[O<n«C~:-c )/cxyJ ,~hich
- 0
For each
minimizes
the optimal value of n is that one (of the three)· yielding the minimum
value of
~ases
V
V(l1y
where
Note that for each of the three
H
(1) U and V have the forms U =
listed above:
= ~ 3+.~.n
2
n J.
IQy tQ x ) = U/[UV
~
t'" t ~
D~O
1
+"~
2
nand
are not functions of n; and (2) we may
1 " ..
consider separately the
~
(a) those values of n where
sub-cases:
W >0, and (b) those values of n where W <0.
n
Then t for each of the sub-cases
n
of cases I and II the restrictions on n are of the form 0 < n
"1here n
L
and n
U
denote the 1'0Hel~
< n ~ n
U
L
and upper bounds for n determined from
the appropriate range of Hand 0 < n <
n
-
-
(C~:-c )/c xy ,
0
Thus t
for each
sub-case (of cases I and II) we find the optimal value of" n by rrJrtimizing
g(n)
= (~ 1+~2 n)/[(~ 1
+~ n)(~ +~ n)
234
Of course,
for each sub-case.
NOH,
~
1
,
~
2
,
~
3
,
~
- (~ +~ n)2J
S6
"Jill assum8 different values
II
(4.10
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g(n) = (v
+ v n)/(v n 2 + v n + v )
2
1
3
where v =~ , v =~ , v =~ ~ -~
4
5
v=~ ~ +~ ~ -2~ ~ , and v =~ ~ _~ 2.
2,
11223246
4142356
5135
The optimal value of n can be determined by examining dg(n)/dn over the
The solution depends on the signs of v ,v
and v :
123
For case I, it can be shown that v
negative or zero.
a~
< i<'O) and v
2
< 0, v
3 -
> 0 and v may be positive,
1
2
Similarly, for case II, v
> O.
> 0 (v
< 0, v
> 0 if
311
Thus, we present beloH the best choice for n
correspon ding to each of four specifications of v
1'
v
2
and v
3
.
These
v
= 0
specifications include all of the situations mentioned above:
(A) v
3
< 0, v
1
-> 0,
v
> 0,
2
1'
v
(B)
v ), (D) v
(any values for v
(A)
2
3
3
< 0, v
< 0, v
1
1
> 0, v
> 0, v
2
=
2
< 0,
( C)
3
o.
v <0, v >0, v >0:
3
1-, 2
(1)
If v v
> v v , take n=n •
L
1 ..
(2)
If v v
< v
2 5
2 5
V
1 4'
take
n=n~':
if n
L
< n:': < n
n=n L if
n,,;':
< n
\-lhere n:': = -(v /v ) + [(v Iv
)2
+ {(v
U
L
n=n if n U < n:':
U
1
12
12
V
25
-.v
v )/v y
lit
2~
}J2
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37
(B)
(2)
If [n* + (v
1
Iv 2 )J2
take
> 0, and -[2(v
n~nL
(3)
If [n~':
-I-
(v
1
Iv 2 )J2
!
•
if n*>n U
> 0, and -[2(v
n =n U
1
Iv 2 )
+ n~':J > nL'
if n*<n
-
u'
-.
and
g{-[2(v
n=-[2(v
1
. (C)
V
3
=0:
If v v <v v , take n=n U'
2 5- 1 Ii
If v v >v v , take n=n ,
L
2 5
1 Ii
( D)
v <0, v >0, v =0:
312
1
Iv
2
)+n*J}>g(n u)
Iv 2 )+n*J
if n*<n , and
- u
g {-[2(v Iv )+n*J}<g(n u)'
,
j
I
L'
take n::min[n ,-{2( v /v )+n*}Jif n*>n U
u
2
1
i
i
+ n*J < n
n=n L if n*~nU' and g(nL)~g(nU)
I
j
Iv 2 )
n=nu if n*~U' and g(nL»g(n U)
I
,
1
1
2
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38
< -(v /2v ) t take n=n •
U
4
3
If n > -(v /2v )t take n=n •
L
L
4
3
If n < -(v /2v ) < n t take n=-(v /2v ).
L
4
3
- U
3
"
If n
(1)
( 2)
( 3)
U
-
For case III, it is easily shown that
To find the value of n [0 < n < (C*-c )/c
.
-
-
0
XY
+ K n)-l
] which minimizes (K
1
2
I < c - c 2 Vn and Ic 2 Ic 1 IWn I -> Un (see (4.10»t
n - n
one may consider the follrn1ing two cases separately: (a) those values of
in (4.13) subject to ;C-C-/W
1 2
n. whel'e Wn>Ot and (b) those values of n where W <0.
n
In~!.:.
is easily seen that the optimal choice for n is given by:
case; i t
If K <Ot take
2
Here t n L and n U are~ respectivelYt the smallest
and largest values of n \-li thin' the range defined by ICCI H I < C - C V ,
n=nL ; if K >Ot take n=n U'
2
1 2
{c
. ·'2
Ic 1 IW n I
> U t and 0 < n < (C*-c )/c
n
-
-
0
XY
•
n
n
2
n
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39
5.
Conclusion
. From the numerical evidence cited in Section 3.3, the estimators
fiYBl~ and
f\13P
are seen to be
credible
alternatives to the estimator
defined by the preliminary test procedure,
lihood (regression) estimator,
fl yR '
fly,
and the Iaaximum like-
SinGe these estimators have been
compared using mean square error as the criterion, the results are even
more surprising.
Of course, !\B1~ and 0YBP have larger mean square
errors than those of
Oy
from the true situation.
sign.
and
fl YR
when the prior assessment is "far"
In particular, this distance seems to be a
HOivever, our numerical results suggest that even when the
prior assessment is
poor, the Bayesian estimators 'dill genex>ally
have the smaller mean square errors.
A second advantage for the Bayesian approach is the ease in
deter-mining the optimal values of the sample sizes n, n
X
and n y '
This
is a very formidable task if the preliminary test approach is employed.
finally, one might expect that the case of E unknoHn could be
handled easily ...l ithin the Bayesian framework.
To indicate the nature
of the problem, consider only a random sample of size n from the
bivariate normal distribution, N (ll,E) (or, e, quivalentiy , a random
2--
sample from the multivariate normal distribution).
for
s~mplicity,
let
2
p
0
0
l1 y =P1' PX::
• 2 " Oy2:: 11 'OX:: 22 ,OXy=O 12 •
Evans (1965) and others have assigned the natural conjugate prior
distribution to (l!..,E); Le •., l!..1l.:-I)2(H.,,~/A), and .E,-l_H(n,p-l) ...,here H
denotcsthe iHsbat't distribution and 'A>O, n>O are scalars.
(19G5) shaHs that
Then Evans
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( 5.l)
where -x is the vector of sample means t and
matrix.
2
the sample covariance
lineal~
Note that the posterior expected value. of 11 1 is a
x1 and lJ'1
bination of
11
~
is not used.
that n=l1 -lJ
1
2
and that the sample (and prior) information about
This appears to be contrary to
o~r
is in the neighborhood of 0t and that 11
"closely related."
com-
Taking
l1
strong prior belief
1
and 11
2
are
=n =o in (2.12) and then comparing
X y
E(llyIQytQ ) with E(!!.lx,.§.) from (5.1) suggests that the natural conjugate
x
prior distribution may not be appropriate for our situation.
This
conclusion is also suggested by the results that the marginal prior
covariance matrix of
~
is proportional to
of E- 1 is proporti~nal to
£.-1.
prior covariancernatrix of
£. while the prior e;<pectation
(Note that if E is diagonal t then the
E. Hill
also be diagonal Hhich may be an
unrealistic prior assessment.)
Geisser (l965a) and others have taken a joint locally uniform prior
(~t~).
ClearlYt this does not accurately reflect our
prior qpinions about P.
Alternatively, Geisser (l965b) has considered
X-N (lJ,L) Hith lJ=(11,lJ).
Th~l)
..distpibution for
-
.2 -
-
-
prior to 11 (scalar) and E- 1 •
he assigns a jointly local1y uniform
This specification is, again, inappropriate
because He are, in general t unHilling to assert (with certainty) that
11 =11 =lJ.
1
2
One might assign a locally uniform prior distribution to It and
take
III E ...
--
N (p', V' ).
2 -
Unfortunately, the posterior distribution of _P
-
does not appear to be tractable. because it "is the product of a bivariate
student distribution and a bivariate nonnal distribution.
of the
last-m~ntioned
Modifications
joint prior distribution do not seem to yield any
more promising results.
I.
41
(,1
1
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Acknowledgments
The' authors wish to thank the National Institutes
of Health, the National Institute of Mental Health, and
the U.S. Office of Education for partial support of this
research.
At the University of North Carolina this
support was granted through the National Institute of
Mental Health Training Grant MH10373; at Iowa State
University this support was granted by the U.S. Office
of Education under contract number OEC-3-o02041-2041
and through the National Institutes of Health Biometry
Training Grant 5TIGM34.
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References
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b~va:i~te normal distribution using a preliminary test of
s~gDJ.fJ.cance and a two stage sampling scheme " Institute
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Carolina at Chapel Hill.
Cochran, W. G. (1963). Sampling !echniques.
York: John Hiley.
Second Edition.
Ne'"
Evans, 1. G. (1965). "Bayesian estimation of parameters of a multivariate normal distribution," ,Tour. Royal Stat. Soc. B, 27,
279-283.
Geisser, S. (l965a). "Bayesian estiT:1ation in multival'iate analysis,"
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Geisser, S. (l965b). "A Bayes approach for combining correlated
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Mosteller, F.
~,
(191~8).
"On pooling data," Jour. Amer. Stat. Assoc.,
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Pratt, J., Raiffa, H. and Schlaifer, R. (1965). Introduction to
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tI)-rc-::-
Roberts, H. (1966). "Statistical dop;ma: one response to a challenge,"
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Ruhl, D. (1967). Preliminary test procedures and Bayesian procedures
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