•
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. c.
AFOSR Report No.
A MIN]MUM AVERAGE RISK SOLUTION FOR THE PROBLEM
OF CHOOSING THE LARGEST MEAN
By
Richard Park Bland
April, 1961
..
Contract No. AF 49(638)-929
The problem of choosing the largest of n means is
considered as a multiple decision problem which is
generated from n component two-decision problems.
With additive losses Bayes rules for the component
problems yield Bayes rules for the multiple decision problem. Some properties of these Bayes rules
are found. Also a conservative-near-Bayes rule is
presented with tabled values for any number of means.
Qualified requestors may obtain copies of this report from the
ASTIA Document Service Center, Arlington Hall Station, Arlington
12, Virginia. Department of Defense contractors must be established for ASTIA services, or have their "need-to-knowll certified by the cognizant military agency of their project or contract.
.e
Institute of Statistics
Mimeograph Series No. 280
ii
•
ACKNOWLEDGEMENTS
This short note is to acknowledge encouragement, buidance
and other forms of help from various sources.
To attempt any
listing except the most fragmentary would be foolhardy,so the
following is by no means complete.
Of primary importance is
Dr. David Duncan for suggesting the problem and having patience
with my feeble attempts at a solution.
Dr. Wm. J. Hall made
many valuable suggestions and criticisms.
The National Science
Foundation, Office of Naval Research and Air Force Office of
Scientific Research were kind enough to grant financial aid
•
at different stages of the preparation for this work.
The
unpleasant task of typing the manuscript fell upon Mrs. Patricia
Ziolkowski and I am grateful for her pleasant execution of this
chore.
To those who know Martha Jordan I need not explain how
helpful she has been in the interpretation of and guidance
through administrative regulations; to those who do not know
her, words cannot convey the debt.
iii
TABLE OF CONTENTS
PAGE
CHAPTER
ACKNOWLEDGEMENTS .
I
II
it
INTRODUCTION .
1
THE PROBLEIvI OF CHOOSING THE LARGEST MEAN
7
2.1
7
Statement of the problem.
2.1a The observed data and decision system
2.1b Loss functions
10
2.1c Prior distributions.
12
2.2
•
III
7
The p-decision problem as the restricted
product of n two-decision problems.
12
2.2a Generation of the p-decision subsets
12
2.2b Generation of the p-decision problem loss
functions.
16
2.2c Generation of p-decision procedures as the
intersection of n two-decision component procedures .
18
2.,
21
Invariance of decision
p~ocedure.
2.4 SUJ::lInary of, ,assumptions
21
2.5
23
Bayes decision procedure.
BAYES DECISION PROCEDURE FOR THE CASE n=3, (j2 KNOWN 29
o
Figure 1 Bayes deciston procedure critical regions
(n=3, (j2 known)(third component problem).
•
o
39
Figure 2 Sample space regions for Bayes decision
procedure (n=3, 0"2 known).
o
43
iv
PAGE
CHAPTER
Table 1 Bayes decision procedure t values,
n
= 3, cro2 known
7
7
7
7
2
= 00
45
2
=3
46
2
=1
47
2
= 0.5 .
48
Table 2 Bayes decision procedure t values, n
=2
2
7=00,3
= 1, 0.5
50
SOME DERIVED DECISION PROCEDURES AND THEIR PROPERTIES
51
4.1 Progressive method for generating a class of
decision procedures for the p-decision problem
51
7
IV
•
2
49
4.2
Some properties of the generated decision
procedures
54
59
4.3 The decision procedures Di (B2 ) and D(B2 ).
67
4.4 Comparison of risk functions of D(B) and D(B2 ) 70
V
Figure 7 Ratio of risk for conservative-near-Bayes
decision procedure to risk for Bayes decision pro2
cedure (n = 3, cro known)
73
THE EFFECT OF DEPARTURES FROM NORMALITY IN THE
PRIOR DISTRIBUTION
74
5.1
Introduction.
74
5.2
Choice of a family of prior distributions
74
5.3 Parameter values used.
76
v
CHAPTER
PAGE
Table 3
77
4 values used
~ > 0:
Table 4 Parameter values used in nonnormal prior distributions
79
Figure 8 Non-normal prior distributions
80
5.4 Bayes decision procedure .
83
Table 5 Bayes decision procedure t values nonnormal prior distribution, n=2,
BIBLIOGRAPHY
•
•
~
2
o
known
90
91
CHAPTER I
INTRODUCTION
It has been generally recognized for many years that the
classical treatment of the problem of comparison of means by a test
of the hypothesis that all the means are equal does not provide
the answers which the experimenter usually desires.
As a result
much work has been done on a class of problems which are felt to
be more applicable.
For example, in some situations an experimen-
ter may conduct an experiment to choose from a group of populations
the one with the largest mean.
Usually, in the interests of not
missing the population with the largest mean, he would be willing
to choose a group of one or more populations such that anyone of
•
them could have the largest mean.
This he might term the superior
group of populations as distinct from the group of populations not
chosen which could be called the inferior group.
He would reason
that due to experimental error it is quite possible for any of the
populations in the superior group to be the population with the
largest mean so he would treat these populations equally and either
do further experimenting to choose between them or decide on the
basis of some other criteria which one population to choose.
Thus
in this case a statistical problem would be to find a fixed sample
size decision procedure for dividing a group of n populations into
a superior group, which can contain from one to n of the populations,
•
and an inferior group containing the remaining populations •
2
It is this decision problem that We study in this paper.
We will
refer to problems of this type as problems of the largest mean.
In other situations an experimenter may conduct an experiment
not to determine just the population with the largest mean but
to determine the complete ranking of all the population means.
We will refer to problems of this type as problems of ranking
means.
With the development by Wald
ffiTJ *of
a general theory of
statistical decisions attempts have been made to find decision
procedures for problems of the above types in the framework of
this theory but it has usually been necessary to use a simplified
approach in order to obtain a solution.
Mosteller LI~7 considered
a 'slippage' problem where it is desired to find which one, if any,
of k populations has a larger value of a location parameter than
the other populations.
The decision procedure proposed was not
claimed to be optimum but only easy and rapid to apply.
Paulson
L1~7 considered the same multiple decision problem as considered in
this paper, that of dividing a set of populations into a llsuperior"
and an "inferior" group.
The decision procedure proposed was only
claimed to be qUite reasonable on an intuitive basis.
In
LI27
Paulson considered the k-decision problem of finding the 'best' of
k populations when k-l experimental populations are compared with
a control population.
*The
A decision procedure was obtained which
numbers in square brackets refer to the bibliography.
3
had a probability
..
~
1 - a of selecting the control population
when it is the superior population.
He also determined the
sample size needed so that the probability of selecting a superior experimental population as best is > 1 -~.
In
1:227
Paulson gave a decision procedure for the slippage problem of
Mosteller that had optimum properties.
Bahadur~!7 considered
the same decision problem as this paper and showed for a particular type of loss function what the best decision procedure
was in the class of impartial decision procedures.
Bechhofer
~g7 considered the determination of sample size for a single
sample experiment so as to make decisions about various rankings
of means so that under specified conditions the proportion of
correct statements associated with the decision procedure 1s
equal or greater than a preassigned value.
Dunnett ~§.7 con-
sidered the same problem for the case where the means have prior
distributions.
In
[""17
Bechhofer, Dunnett and Sobel gave a two
sample decision procedure for ranking means similar in method to
Bechhofer's earlier single sample decision procedure.
Bechhofer
and Sobel ~~7 considered a sequential decision procedure for the
problem of ranking means.
Hall
LI!7
formulated the same problems
as considered by Bechhofer, Dunnett and Sobel
LY,Li!,
~y
but, instead of placing a guarantee on the probability of making
the correct choice, placed a guarantee on the probability of
making a distinctly erroneous choice.
Seal L2~7, working with
Bose, gave a general class of decision procedures for the same
4
decision problem as this paper and showed that the members of
this class possess some desirable properties.
Gupta
L127,
another student of Bose, considered one decision procedure from
the class considered by Seal and showed it has some other desirable properties.
Decision procedures for the problem of
ranking means have also been proposed by Fisher ~27, Newman
617, Duncan L'2.7, L§7 and Tukey ffi'2.7, ffi§},
L2L7 proposed a decision procedure for making
while Scheffe
comparisons be-
tween all contrasts of the means.
In all of the above the approach has either been to find
the properties of some
pro~osed
decision procedure or to find a
decision procedure which has some specified good properties.
None of them have given a complete treatment in terms of Waldls
theory.
However in ~17 Duncan formulated the problem of
ranking means as a pairwise-multiple comparisons problem in
terms of Waldls theory and, for a particular type of loss
function and prior distribution of the means, found a Bayes
(or minimum average risk) decision procedure.
The present
paper considers the problem of choosing the population with the
largest mean from a group of n normal populations in terms of
Waldls theory as a multiple decision problem and, for a particular type of loss function and prior distribution of the
means, presents a Bayes decision procedure.
A key development
is that of generating the multiple decision problem involved
as the restricted product (Lehmann
[ig7, lJ-L7)
of n simple
5
two-decision problems.
In ~17 Duncan used a similar technique
in dealing with the problem of ranking the means.
Also both of
these techniques are similar to the 'union-intersection' principle of test construction proposed by Roy
L2g7.
In Chapter II the multiple decision problem to be studied
together with the loss functions and prior distributions of the
means are given.
Then it is shown that Bayes decision procedures
for some two-decision 'component' problems generate Bayes decision procedures for the original multiple decision problem.
The
Bayes decision procedures for these 'component' problems are
given only implicitly with the final solution depending on the
evaluation of certain integrals.
In Chapter III numerical values are given for this decision procedure for the particular case where there are three
normal populations with a common and known variance.
Tables
are provided to facilitate the use of this decision procedure.
For the case where there are two populations the pairwise-multiple comparisons problem as formulated by Duncan ~17 and the
problem of choosing the largest mean as formulated here are exactly the same so Duncan's results provide the Bayes decision procedure for our problem in this beginning special case.
of Duncan's results is included.
The table
Also it is noted that in con-
trast to previous approaches to the problem of choosing the
largest mean the present approach leads to 'significant differences' that decrease with the number of populations included
6
in the problclm.
In Chapter IV we find) b J
~he
of test construction proposed by Ro,Y
union-in~ersec~ion
ffi.?-7,
method
a class of d,;cisioli
procedures for our original multiple decision problem.
These
decision procedures are not Bayes but are 'conservatively-near'
Bayes (in a sense given in Chapter IV).
Some interesting re-
lations between these decision procedures and the Bayes decision procedure are given.
In particular it is shown that the
decreasing-with-n nature of the Bayes decision procedure noted
in Chapter III holds in general
Also the ratio of the risk of
one particularly simple decision procedure to the risk of the
Bayes decision procedure is presented for the case of three
normal populations with a common and known variance.
Chapter V presents a study of the variation of the Bayes
decision procedure with changes in the prior distribution of
the means.
This is done for the case of two populations so it
is applicable not only to the problem considered in this paper
but also to the problem of ranking means as formulated by
Duncan
["17.
CHAPTER II
THE PROBllivl OF CHOO::;ll\TG THE LARGEST
2.1
lvIEAl~
Statement of the problem.
2.1a The ob served data and d·ecision f3ystem.
Let nl,n , ... ,n be n normal populations witb means
2
n
2
.,~n and a common variance ~0 ,where -~=(~l>" .,~n )
2
~1/~01"
_ ~
is unknown and ~
o
is either known (case 1) or unknown (case 2).
Suppose there are m random observations x .. (.j = 1, ... ,m) from
1,1
each of the n normal populations
~.
(i = l, ... ,n).
J.
denote the set of the n ~'s J S = {~l"'" ~ n }
~.
{
Jl
j
~.
J2
,
••• ,
~.}
(Let S
and
denote the complementary set
Jk
With the information obtained from the nm random observations
we consider the problem of finding a Bayes decision procedure
n
for choosing one of the p = 2 _ 1 decisions:
d.
1
:
~ €
-
UJ.
1
,
(i = l,2, ... ,p);
t.he p = 2: Pj subsets w. being defined as follows:
1
p
the first
n subsets are those in which one specified mean ex= (n)
1 =
1
ceeds the remaining n-l means,
Jj
p
8
•
the next P2== (~) == n~/2~ (n-2)~ su1f;ets /:ire those in which l:he
lar~er
of two specified means exceeds the remaining n-2 meal,s)
W'P +l == {!:j max {Ill' f.l 2 } > max \ Ill' 112 \ }
1
l}
w +n - l == {!:j max {f.ll,f.lnt > max {Il l ,f.ln
Pl
w +P == {!:j max {f.ln_l,f.ln\
Pl 2
,. . .
,.
> max {Iln _ l ,f.ln )}
the next P == (~) == n~/3~(n-3)~ subsets are those in which the
3
largest of three specified means exceeds the remaining n-3 means,
W
Pl
+P +l== {!:j max
2
{f.ll,~12,1J.31 > max ~~1}
t
wp +P +n-2==\!:; max {f.l l ,f.l2 , lln
1
2
> max {f.l l ,f.l2 ,f.ln
,...
1}
1 > max {f.l l ,f.l3'f.l41J ,..
w +P2+n - l == {!:; max {ll l ,f.l3'f.l4
Pl
the next to the last P - == (n~l) == n subsets are those in which
n l
the largest of (n-l) specified means exceeds the remaining mean,
the last P == ( n) == 1 subset is the set in which anyone of the n
n
n
means may be the largest of all the means,
W
p
= fl
(the entire parameter space of ~).
9
These p decisions can ue viewed as decisions clas6ifying the n populations into a superior group, which is to
contain the population with the largest mean) and an inferior
group.
(The terms superior and inferior are as used by
Paulson
LI§7).
From this point of view the p decisions may be
described as:
n decisions specifying exactly one population in
the superior group,
(~)
= n:/2:(n-2}:
decisions specifying exactly
two populations in the superior group,
(n~l)
=n
decisions specifying exactly n-l
populations in the superior group,
and one decision specifying all n populations
in the superior group.
This p-decision system is the same decision system considered previously by Paulson
17,
p. 18
L!§7,
very briefly by Duncan
Bahadur ~17, Seal L2~7, Gupta
For n
= 3,
:!:
€
,
d
3
!:
€
(l)2
=7
decisions:
t!:i III > max (1l2 ,1l3 }} ,
(l)l=
that is,
d2
others.
the case considered in more detail 1n the
next chapter, there are the following p
dl
LI27 and
~l
= {!:j
is the superior group,
11
2
> max (ll l ,1l }}
3
that is,
~2
!:
{l:i 11 > max (1l1 ,1l2 )} ,
3
€
(l)3 =
that is,
~3
is the superior group,
is the superior group,
£5,
10
d4
~
€
~
d
~
€
~ = {~j max (I-l l , 1-l ) > 1-l2 } ,
3
= {!:;
max (I-l l , 1-l2 ) > 1-l } ,
3
that is, n and n are the superior group ,
l
2
5
that is, n
d
d
are the
3
~ € ill6 = {~; max (1-l2 ,1-l ) >
3
that is, n and n are the
2
3
6
and n
superior group
J
I-l }
l
superior group ,
n
~€~=
7
l
that is, n ,n and n are the supertor group.
l 2
3
2.1b
Loss functions.
For anyone of the p decisions the loss function is defined
in the following manner:
let I-lmax= max (l-ll, •.. ,l-l ),
n
let d be the decision that
n. , ... ,n
J2
n.
J k +2
jk
, ... ,n j
are the superior group
J
are the inferior group,
n
(Where (j1, ..• ,jn) is a permutation of (l, •.. ,n»,
that is, d is the decision that
I-l
-
€
{I-l; max (I-l. , ... ,I-l. ) > max(l-l
-
J1
Jk
j k+1
then the loss incurred by d at any point
, ... ,I-l.)},
In
!: is
k
L(d;~)
=E
i=l
(k /rr)(1-l
- I-l. )
0
max
Ji
n-k
+ E (kl /2CY) [JI-l.
- max{l-l j
\)+ II-l.
- max{l-l. 1/7.
i=l
Jk+i
k+i
J k +i
J k +i -
11
where k > k > 0 are loss proportionality factors,
l
o
and
=
CJ
CJ
o
/fTi .
That is, the total loss for a
~iven
decision is the sum of the
losses made with respect to each individual population.
contribution of n
i
to the loss is as follows:
if n
ded in the superior group a zero loss is made if
a positive loss proportional to
(~
max
-
~.)
~
i
The
is inclu-
~.= ~
~
max
is made if
and
~i< ~
max
;
if n. is included in the inferior group a zero loss is made
~
if ~.~ < ~max·and a positive loss proportional to (~i- max {~.}
)
~
is made if
~i= ~
max
.
To illustrate with a simple example:
n = 3 suppose decision d
2
for
is made when ~ = (5, 7, 10) where
decision d2 is to state ~ € (1.)2= {~i ~2 > max(~l' ~3)}' that
is, n is the superior group. Then the loss is:
2
L( d
2
i
5,7,10) = (ko/CJ)(lO - 7)
+ (kl /2CJ)LT5 - 10) + 15 - 10 1_7
+ (kl /2CJ)L{10 - 7)+ 110 - 71_7
= 3(k /CJ) + 0 + 3(k /CJ) .
o
l
This includes a loss of 3(k /CJ) for incorrectly retaining n
o
2
in the superior group, a zero loss for correctly placing n
..
l
in the inferior group and a loss of 3(k /CJ) for incorrectly
l
placing n in the inferior group.
3
As will be seen only the ratio kl/k = k is needed to be
o
known and not k and k • This ratio will be called the loss
o
l
ratio or error-seriousness ratio (terms first used by DuncanL-17
12
for a similar ratio).
(As seen later we must require that k
be not less than three
f~r
2.1c
the case n
= 3.)
Prior distributions.
As the final assumption needed for a Bayes formulation
of this p-decision problem the
~'s
are given independent
identical normal prior distributions, each with an unknown
22222
mean.c' and variance r (J where r is known. (J = (J 1m is
o
taken as fixed and hence has no prior distribution.
While it is not difficult to formulate this problem
as above, in the framework of the general theory of statistical decisions as developed by Wald, a direct approach to
find a Bayes solution is very difficult.
Others in attempting
to surmount this difficulty have used simplified approaches to
find decision procedures for the p-decision system that have
some desirable properties.
However it will be shown in the
following section that with an indirect approach we will be
able to find a Bayes decision procedure without the need of
any further simplifications.
2.2
The p-decision problem as the restricted product
of n two-decision problems.
2.2a Generation of the p decision subsets.
The subsets of the p-decision problem as formulated in
the previous section can be generated as the restricted product (a term introduced by Lehmann
LIgl,
ilL?
to be described
below) of the subsets of n two-decision component problems.
13
For this purpose, let us consider a two-decision problem
(which will be called the i
th
component two-decision pro-
blem) where the pair of subsets are:
W
io
; 1l , the entire parameter space of
~,
and
W
il
; {~; Il < Il
} , that is, in w some Il j
max
i
il
is larger than Il .
i
The corresponding decisions for this component problem may
be denoted by:
Il e w '
il
and
Viewed as decisions to classify the
populations into a superior group and an inferior group, where
initially all the populations are considered as being in the
superior group, decision d
iO
is the decision to leave
superior group while decision d
il
fi
i
in the
is the decision to place
fi
i
in the inferior group.
= 1,2, •.• ,n
For i
there are n such two-decision problems.
Now if these problems are considered simultaneously, they generate a multiple decision problem where the different decisions
are statements about
~
being in the intersection of the n com-
ponent subsets.
Thus if (jl, ..• ,J ) is a permutation of
n
i
(l, •.• ,n) and if 1n the J. th ) .•. ,J.th component problems dec i sons
l
k
· the Jk+l,
.th •.• ,J. t h
.
t procomponen.
d. , ... ,d. 0 are made and ~n
n
J10
J
k
blems decisions d.
J k +l
l, ..• ,d. 1 are made then in the multiple
In
decision problem the decision that corresponds to these
14
component decisions being made simultaneously is
k
n
:; : i=l
n m.J i 0 i=k+l
n m.J 1
i
that is,
n
dk : !:
€
~ = .ll ()
i=k+l
m. 1 (since (J)iO= 11. for all i).
Ji
Viewed as a decision to classify the populations into a superior
and an inferior group, this decision classifies
~.
, ... ,~.
Jl
""'~j
in the inferior group.
k+l
n
Not all of the products of the component subsets form
in the
sup~rior
J 1<:
6rouP' and
~j
decisions in the p-decision problem. If d
is the decision
il
th
made in the i
component problem (i=l, ... ,n) then considered
simultaneously this is a decision to assert
n
1J.€(\m
. 1 il
~=
But since
nmil=
·
0 this decision represents an impossibility
and is not desired in the product problem.
(This decision says
for each and every mean that there is another mean which is
larger.)
Therefore in general a product decision problem is
restricted to dec1a1cI:S about nonempty subsets.
1127
Lehmann
[ig7,
refers to such a generation of a multiple decision problem
from a set of two-decision problems as a restricted product
decision problem.
More specifically the subset system generated
by the nonempty intersections of the component subsets is
termed the restricted product of the component subset systems.
n
In the case under consideration only one ( (\ mil) of the
~=l
15
component subset intersections is empty.
The rest all form
elements in the p-decision system.
To illustrate the foregoing, for n=3 there are three
component two-decision problems with the following decision
pairs:
for the first component problem
that is, leave n
dl l
!:
€
(J.)ll
= {!:;
l
j..Ll
that is, place n
l
in the superior group,
< max (j..L2,j..L3)}
in the inferior group,
for the second component problem
that is, leave n
d 21
!:
€
(J.)21 =
{!:;
2
in the superior group,
j..L2 < max (j..Ll,j..L3)} ,
that is, place n
2
in the inferior group,
for the third component problem
that is, leave n
d 31
!:
€
(J.)31
= {!:;
3
in the superior group,
j..L3 < max (j..Ll,j..L2)}
that is, leave n in the inferior group.
3
The p = 7 decisions in the restricted product decision problem
are generated as follows:
16
d l = (d lO ' d21 , d ): ~
31
€
U). = wlO
n w21 n w31 '
d2
= (dll ,
d20 , d ): ~
31
€
w2
= wll n w20 n ~l'
d
3
= (d ll ,
d 21 , d
): ~
€
w
3
= wll n w21 n w30 '
d4
= (d lO '
d 20 ,
d31 ): ~
€
w4
= wlO n w20 n w31 '
d
5
= (d lO '
d21 , d30 ): ~
€
~
= wlO n w2l n
d6
= (dll ,
d20 , d30 ): ~
€
w6
= ~l n w20 n ~O'
d7
= (d lO '
d 20 , d30 ): ~ e
30
w30 '
w.., = wlO n w20 n w30 '
The remaining product decision
states that
~
is in an empty set hence the product decision
problem is restricted so as not to include this decision.
With this method of generating the decision system of
the original p-decision problem as the restricted product of
n two-decision problems and, with the loss functions for these
component problems as given below, a considerable reduction is
made in the difficulty of the sUbsequent derivation.
2.2b Generation of the p-decision problem loss functions.
For the i
th
of the two-decision component problems let
loss functions be defined as follows:
L(d
L(d
L(d
iO
; ~)
= 0,
iO
; ~)
= (k0 /~)(~max-
-~ e w' l '
il
; ~)
= 0,
~ € wil'
-
L(d il ; ~)
= (kl/~)~i-
~ €
_
•.1.1..
1\
-
w'~ l '
~
max(~1""'~i_l'~i+l""'~n17,~ €jL- wil'
Thus for the i
th
component two-decision problem, a zero loss
is made if decision d
or if decision d
A
il
iO
is made when
is made when
~i
~i
is the largest mean
is not the largest mean.
positive loss proportional to the difference between the
largest mean and
is made when ~i
iO
is not the largest mean and a positive 1066 proportional to
~i
is made if decision d
the difference between
~i
and the next largest mean is made
if decision d
i6 made when ~i is the largest mean. That is,
il
a zero loss is made if n. is put in the correct group while a
l.
positive loss is made if n. is left in the superior group when
l.
e
~i
< ~max or if ni is placed in the inferior group when ~.l. = ~max .
With these loss functions for the component problems it
is
seen that the loss functions for each of the decisions of
the restricted product decision problem is the sum of the loss
functions for the decisions of the component two-decision problems that generate that particular decision.
That is, if d
i
is a decision of the restricted product decision problem and
dlil,···,dnin are the n decisions from the n two-decision
problems (one from each problem, i = 0 or 1) such that
k
d
i
is (d
li 1
, ••. ,d . )
nl.n
than
L(d ; 1:)
i
=
n
E
k=l
L(d · ; 1:) ,
k l.k
This will be referred to as additive 106S.
To illustrate let us refer to the example used previously
18
for the case n
IJ. € ill
=
[1:;
1J.
2
2
(The decision d
= 3,
with decision d :
2
> max (Ill' 113)} being made when ~ = (5, 7, 10).
is to leave n in the superior group and place
2
2
in the inferior group.) As seen before, the loss for
n and n
l
3
this particular example is:
in the p = 7 decision problem results from making
2
the component decisions d , d , d
simultaneously. The losses
20
ll
3l
for these component decisions are:
The decision d
L(d
ll
; ~)
=0
L(d 20 ; !:)
= (k0 /J)( 10
; ~)
= (kl / J )( 10
L(d
31
-
7) = 3(k0 /J)
7)
= 3 (kl/J) ,
which add to the given loss for d . For the first component
2
a zero loss is made when n is placed in the inferior group
l
since III is not the largest mean; for the second component a
positive loss of 3(k /J) is made by leaving n in the superior
o
2
group since 112 is 3 less than the largest mean and for the
third component a positive loss of 3(k /J) is made by not
l
leaving n in the superior group since 113 is the largest mean
3
and is 3 larger than the next smaller mean.
2.2c
Generation of p-decision procedures as the intersection
of n two-decision component procedures.
If for each component problem we have a decision procedure
such that considered simultaneously these decision procedures
19
do not lead to any inconsistencies, that is, they do not make
the statement that
~
is in an empty set, then they form a com-
patible set of decision procedures.
As proved by Lehmann
LTg?,
decision procedures for the
restricted product decision problem obtained as the intersection of a family of compatible decision procedures for the component decision problems are in one-to-one correspondence with
the generating families of compatible decision procedures for
the component decision problems.
That is, a set of compatible
decision procedures for the component decision problems generates
a decision procedure for the restricted product decision problem.
And since the decisions of the restricted product problem are the
same as the decisions of the original p-decision problem, it
follows that a set of compatible decision procedures for the component decision problems generates a decision procedure for the
original p-decision problem.
To insure the compatibility of the n two-decision procedures
the condition
must be met.
That is, each of the two-decision procedures must
be chosen in such a way that when considered simultaneously the
probability of placing all the populations in the inferior group
is zero.
Following Lehmann
LTg?
this will be referred to as the
condition of compatibility for the component decision procedures.
Not only does a family of compatible decision procedures
for the component two-decision problems generate a decision
20
procedure for the restricted product problem but, with the
same prior distribution for the
IllS
in the component problems
as assumed in the original p-decision problem and with additive loss, a family of compatible Bayes decision procedures
for the component problems generates a Bayes decision procedure for the restricted product decision problem.
This most
useful and important result is a simple consequence of the
fact that with additive loss the risk is additive and, since
expectation is a linear operator, the average risk is additive.
Hence if each component of the Bum of average risks is minimized then the sum is minimized.
Therefore minimum-average-
risk decision procedures, that is, Bayes decision procedures
for the component two-decision problems generate a minimumaverage-risk decision procedure for the restricted product decision problem.
(A similar result was noted and used in another
problem by Duncan ~17 and the general result for any restricted
product decision problem was proved by Lehmann
LIg7.)
Thus to obtain a Bayes decision procedure for the pdecision problem we need only to obtain compatible Bayes decision procedures for the n two-decision component problems
and to apply these decision procedures simultaneously.
It is
to be noted that there is complete symmetry among all of the
n two-decision problems (as desired, since we are assuming there
is no prior reason to distinguish between the means of the various
populations).
Therefore a Bayes decision procedure for anyone
of the two-decision problems will yield a Bayes decision procedure
21
for any other, hence for all the others.
Thus the original problem of finding a Bayes decision
procedure for a p-decision problem is reduced to that of finding a Bayes decision procedure for one two-decision problem,
it being understood that the condition of compatibility must
also be satisfied.
2.3
Invariance of decision procedure.
In a.ddition to the requirement that the decision proce-
dure be Bayes with respect to the specified loss functions
and prior distributions we require that it be invariant with
respect to changes in scale and changes in location.
2.4
Summary of assumptions.
All the preceding assumptions are listed below with
comments as to how they could be relaxed.
1.
Fixed, equal size (=m) random samples from n inde-
2
pendent normal populations with a common variance, rr .
o
2.
Additive loss, that is, the loss function for one
of the product decisions is the sum of the loss functions
for its generating decisions.
3.
Loss functions for the individual decisions in the
two-decision problems are proportional to
(~
max
-
~.)
1
diO is made and proportional to
~.- max(~1 , ... ,~.1- l' ~.1+l'···'~n )-
L"' l
if d
il
is made.
7
(Proportionality constant is zero if
if
22
the correct decision is made.)
4.
The population means have independent identical
normal prior distributions, each with mean c' (unknown)
22
and variance I' 2
cr (I' known), (cr
2
5· Case 1, cr0 known.
2
Case 2, cr unknown.
2
2
= cr 1m).
0
0
Assumption 1 can be relaxed somewhat as follows.
From the random samples the set of sufficient statistics
Xl 'X2" "'Xn
tained.
and
s~, estimate of cr~ (if cr~ unknown), are ob-
Thus if we let!
= (Zl""'Zn_l)
be the vector of
n-l orthogonal comparisons in the Helmert transformation
of the
tics
XiS ,
for~'
and if z' and s20 are a set of sufficient statis2
= K~'
and cr then it is necessary to assume only thct
o
z' has a multivariate normal distribution with mean
2
and variance Icr.
i' =
K ~'
Also instead of having equal size samples
from normal populations with a common variance, it is necessary
only that the sample means have a common variance.
With this relaxation of assumption 1 it is necessa.ry to
assume in assumption 4 only that the differences of the
(~.- ~ , i
J
~
f
j, i = 1,2, ... ,n j = 1,2, ...
~IS
,n)
have independent identical normal distributions each with mean 0
and variance 21'
2 2
C1'
•
23
W11at follows is the same illlder either the original or the
relaxed assumptions.
However, for ease of exposition in treating
the more essential features of the problem the original simpler
assumptions will be used.
2.5 Bayes decision procedure.
As shown in the preceding sections the problem of finding
a Bayes decision procedure for the original p-decision problem
is reduced to that of finding a Bayes decision procedure for one
two-decision problem.
For a well-behaved two-decision problem,
such as this one, the method of finding the Bayes decision procedure is fairly straight forward.
A derivation of this method is
next outlined.
Let x
i
= l:
x . . /m be the mean ot' the sample from
1J
(i
and
s
2
o
= the
estimate of
(J'
1(1"
= 1,2, ... ,n)
2
o
Then (-xl, ... ,x,
s 2) is a suffin
cient statistic for the problem.
From the
0
req~irement
of invar-
iance under changes in scale and invariance under changes in 10cation attention may be restricted to decision procedures depending on the random observations through the statistic I
where Z
-
= (Zl""'Zn- 1)
= !/s
is the vector of n-l orthogonal comparisons
in the Helmert transformation
of the XIS and s
= s o /;m .
(In the Helmert transformation,
24
k
=
(1/
in
1/ iii. )
1/ iii.
and
K:;:
1/ j2
-1/12
o
o
fb
1/ Ib
-2/ jb
o
1/
1
1/ In(n-l)
1/ In(n-l)
1/ In(n-l)
L..
2
l*=
.)
. . • -(n-l)/ In{n-l)
jmI is used instead of s in
For the case where cr is known cr :;: cr /
o
0
the above, that is, the statistic
l
~/cr
is used.
(In the following
the symbol X will be used, with the understanding that it would be
replaced by
l*
2
for cro known.)
Also let
~
= 1/cr
where
l
is the vec-
tor of n-l orthogonal comparisons in the Helmert transformation of
the Ills.
Let f(X;
~)
be the probability density function of
X given
g(~)
the prior probability density function
of
L(di;~)
~,
~,
the loss due to decision d. when g is
~
-
the parameter point,
Y
and
sample space of X (or of
parameter space of
r
if cr; known)
~,
(fig is the image of It. under the transformation made from the Ills to the gls).
25
Then for a two-decision problem,
where P(d.; 9) is the probability of making
J.
-
= 0,
the decision d. (i
J.
1) when 9 is the
parameter point.
a(l) where,
Introduce the test function
=
{O, for dO
1, for d
Then P(dO;
~)
~) = ]<1
= Ey(l - a;
-
l
~)
- a) f(I;
dI
Y
r-
and P(d l ; ~) :: EI(a;~)
J
==
f(I;~) d;t.
a
Therefore,
y
risk
~)
= L(dO;
J(l -
=
~) + L(d ; ~)
E1(1 - a;
l
~)
f(l;
~)
f(l;
£)
dl .
a) L(d O;
E1(a; ~)
dI
Y
+~
Y
Average risk
J
=
IL
J
~)
L(d 1 ;
risk
~ (~)
dQ
Q
«1 - a) L(dO;£) + a
L(dl;~)) f(l;~)dl
Y
Then if the order of integration can be changed (as can be done in
the cases herein considered) we obtain:
26
average risk=
j
r
y
J' J'
J J
=
Y
+
L(dO'£)
f(~)£) ~ (~)
d£
d~
-J\.Q
(L(d l ;£) - L(dO;£))
a
Y
__iL
f(X;~) ~ (£)
dQ
d~
Q
Let
then the average risk is minimized by the test:
a = 1 for g(X) < 0 ,
= 0 for g(~) > 0
arbitrary for
g(~)
= 0 .
g(~)
To remove the arbitrariness let a • 0 where
made decision dO if g(~) ~ 0 and make decision d
= O.
l
That is,
if g(z) < O.
Thus for, say, the i th component problem,
diO
~
dil
~
€
gi(~)
=
,J
wiO ' (£
€
w )
i1
€
wiO (£) )
(£ e wil (£) ),
we have
JL g
=
r
J
*
.0~g
L(d
,
- L(d ; ~)
iO
il ~)
f(~, £) ~(£) d9
_I
L(do , g) - L(d. ; 9)
]. O
1. l
-
f(~; £) ~ (~) d9
-
27
where J-lQ*
=.J
~Q
(
- Lpoints
where two or more of the G'S ar'2 r;qual.\,
(this does not affect the results since the ~-measure of the
omitted points is zero).
Therefore
= -(ko /cr)
(where
~.(Q)
~
-
and
~.(Q)
are functions of
J -
the Helmert transformation from the
~'s
_Q given by the inverse of
to the
QI S ,
.)(-
A = ,fLQ () {~; ~j > max {~)}
* where
points in i'L Q
~.= ~
max
J
is the set of
and
is the
Il.~+1'"'' Iln
».
With the function g. the Bayes decision procedure for the i
th
~
component problem is given by the following:
if g.(y) > 0 and make decision d'
~-
-
~
l
make decision diO
if g.(y) < O.
~-
The Bayes deci-
sion procedure for the other component problems follows from this
one, as noted previously, due to the uniform treatment of the Il'S.
Finally, if the condition for compatibility of these component decision procedures is satisfied then the Bayes decision procedure
for the p-decision problem is the product of these n two-decision
procedures.
28
To carry out the necessary integration to permit practical
use of the decision procedure presents some difficulties.
Due
to the non-elementary nature of these integrals this integration
must be done by numerical methods, however, for the case n=3
and ~; known, the tables
LIL?
and
LI§7 have
been used to evaluate
the integrals, and the solution for this case is given in Chapter III.
Cases where n > 3 or n = 3 and ~2 unknown must wait on some numero
ical integration work for the exact procedures. However in
Chapter IV a decision procedure is given that is 'conservative'
and 'near' to the Bayes decision procedure.
For this decision
procedure results are presented that can be applied to practical
situations for all n and
2
known and unknown.
o
~
CHAPTER III
BAYES DECISION PROCEDURE FOR THE CASE n
= 3,
2
cro KNOWN
As in the last chapter
let
gi (X)
=
J
.LL(dil ;
~)
- L(d iO ;
~)_7
f(l'
~) s(~)
d9
*
1\.g
where
I = l* = !/cr since CT~ is known.
As shown in the last chapter g.(y) determines the Bayes
J. -
decision procedure for any of the three (=n) two-decision problems
and, due to the symmetry of these component problems, it is necessary to find the Bayes decision procedure for only one.
Let us
then consider the third component problem with the two decisions
(~
€
(/,)30(~) )
(~
€ (/,)31 (~)
)
For this component problem the loss functions as previously
defined are:
o
, IJ.3 = IJ.max
30
o
, g2 < gl/J3 and g2<
j"3 g2 )/.[2,
-gl/J3
gl
2:
0 and g2
-k1 (gl+/3g2)//2 , gl
2:
0 and g2 ~ -gl//3
k o (91 +
> -gl//5
o
-k1 ( j39 -g )//2
2 1
Thus
g3(1)
= -
oco
j
J
-gl/15
o
-J !
-co
f(l; ~) ~(~) dG 2
Lko (!3g2- g 1)/12_7 f(Z;
£)
~(~)
dg 2
gl/13
(/15
o
d9
-co
g
1J ffi 1 (!3Q2- 1)/12_7
f(l;
~) ~(~)
dg 2
-co
00
J !
-gl/15
dg 1
o
ffiO (gl+139 2 )//2_7
co
dg 1
J
gl::: 0 and g2 ~ gl/13
,,,co
dg 1
o
,
-00
lk1 (gl + 1392 )/12_7 fCV£) ~(~)
dg 2
.
•
31
where
and
Since it is desired to determine the points (Yl , Y2)
where g3(Y ' Y ) ~ 0 positive proportionality terms that will
l
2
cancel out and not affect the final result can be discarded in
this determination.
By completing the squares of 9
1
and 92 in the exponent of
f(Z; 9) g(~) we have (where ~2= (y2+ 1) jy2):
Hence omitting the positive factor
which does not affect the sign of g3' we have:
g3 oc -Il -I2-kI -kI4
3
•
32
-Q/13
co
j'" dQ1j
14 =
o
2
(Q1 + )392 ) D"l¢(Qi; Yi/132,p-l)dQ2'
-co
co
II+ kI 4 = k
00
I f
o
d9 1
(91 +/392 ) IT/(91 ;
-co
y/~2,~-I)d92
Subtract and add the factor
to the integrands in 1 and 1 to obtain:
6
5
co
00
1
5
•
=Jft(Q1o
+f!.,Y1 +
2
y 1 /13 )dQ 1
J....
2
2 -1
E1¢(Qi; Y/13 ,13 )dQ2
-co
l3y2 ) II3~7 j
:f
o
co 2
dQ 1
J
-co
1
lll¢(Qi; Yi lll,f3- ) dQ2
•
33
1
6
= similar expression except lower limit of integration for 9 2 is
In 1 and 1 , let v
6
5
= ~(Ql- Y1/~2)
1
5
= 13-
J
-Y
where ¢(v)
1
6
1
and w
= p(Q2 - Y2/~2)
co
00
1
-91//3.
00
J
v ¢(v) dv j'¢(w) dw + /313-
1
113
-y
-00
= ¢(v;O,l)
1
and ¢(w)
then
00
¢(v)dv j'w¢(w) dw
113
-00
= ¢(w;O,l),
= similar expression except lower limit of integration
for w in each term is (-v/!3)-(Yl+ j3Y2)/(!3f3)
J
00
But
00
¢(w)dw
= 1, and
-00
J'
w¢(w)dw = 0.
Therefore
-00
00
00
1
5
= 13-
1
J
V¢(v)dv + (Yl + l3y2 ) 13-
-yl /f3
2
J
¢(v)dv
-ylli3
= i3-1¢(-Y1/~) + (Yl + I3Y2 )f3-
2
LI - !
(-yl /f317
= j3-1¢(Y /f3) + (Y + !3y ) 13- 2 !(yl /f3),
l
l
2
since ¢(-a) = ¢(a) = (2n)-1/2 exp(_2- 1 a 2 ),
J
a
00
v ¢(v) dv = ¢(a) and
a
1 - ! (-a) = !(a)J where !(a)=j¢(X)dX.
-00
=,!
1
7
00
00
v¢(v)dv
-Yl /f3
J'¢
(w)dw
.
(-v//3)-(Yl +!5y2 )/(!3f3)
Integrate by parts by letting
=v
dr
and
¢(v)dv
00
s
=!¢(W)dW = 1-
~{(-v/l3) -
(-v//5)- (Yl +!3y2 ) /(/313);
(y1 +/3y2 )/(/3i3 )}
v =
00
Therefore
00
J""' ¢(v;
(-Yl -
l3y2 )/4f3
, 13/2)dv.
-Y1 /f3
Let u = (2//3) {v+(Y + !3y )/4i3}
l
2
and the integral in this expression
35
becomes
so
I7
= ¢(yl/p)
~ (Y2/~)
+ (1/2) ¢ {(Yl +j3Y2)/2~}
00
I e=
~ {(!3yl - Y2)/2~}
.
00
j¢(V)
-Yl/~
dv
,!
W
¢(w) dw
(-v/j3)-(Yl+/3Y2)/J3~
00
=
J'
-Y
1
but this is
Let v
=v
¢(v) ¢{C-v//3)- CY1+!3
/p
J3
and u
Y2)/C!3~)} dv,
times the second integral in I ,
7
Therefore
= Cv+/3w)/2;
then
But this is just the integral of a bivariate normal distribution
and can be found using the tables LI~7,
these tables
Using the notation of
36
Collecting and combining the preceding results, we have
I l +kI4
= k~-l ¢(Yl/~)
+ k~-2(Yl+ j3y ) ~(Yl/~)
2
_(k_l)~-l ¢(Yl/~) ~ (Y2/~)
-2(k-l)~-~ {(Y1 + I3Y2)/2~}
! {(j3Yl-Y2)/2~}
- (k-l)~-2(Yl + ./3Y2 ) L( -Yl/~' -(Yl + l3y2 ) /2~, 1/2) .
o
1
2 +kI3
co
= l)~-dQl~Ol+ J302)¢(Gl;-Yl/~2,~-1)¢(Q2;Y2/~2,~-1)dQ2
co
-00
o
- (k-l)
co
1< -dOi)! (Oi+ )302) ¢(oi; -Yl/~2,~-1)¢(02;Y2/p2,~-1)d02'
00
-oi/ /3
Then interchanging the limits of integration for 0i and dropping
the primes,
37
CD
12 +k1
3
00
= k ft,\j<Ql + I3Q2 ) ¢(Ql;-Yl/132 ,p-l)¢(Q2;y2/p2 ,p-l)dQ2
o
-co
00
(X)
- (k-l)
r-Q1J
o
-Q
l
2
(Q l + j3'Q2)¢(Ql;-Yl /13 ,13-1)¢(Q2;y2/tl ,f:3-l)dQ2'
//3
2
but this is the same as 1 +k1 except Y /13 in I +kI4 has been rel
l
4
l
2
placed by -Yl /13 . Therefore using the results for I +kI ,
4
l
hence by combining terms, letting r
= Yl /13
and s
= Y2/13
ting the positive factor (k-1)/13 , we obtain
-1
-1
r:;
g3 ( Yl , Y ) c:c -2k ( k-l )
¢(r) + k(k-l)
(r - ,,3s)
2
- 2k(k-1)-1 r~(r) + 2¢(r) ~ (s)
+ 2¢ {(r + J3s)/2}
!
{(l3r - S)/2}
I3s)/2}
!
{-(/§"r + S)/2}
+ 2¢ {(r -
+ (r + I3s) L(-r, -(r + /3s)/2, 1/2)
- (r - /3s) L(r, (r - /3s)/2, 1/2).
and omit-
Let her, s) be the
exp~ession
on the right.
Thus finally
~e
have
~here
positive proportionality terms have been omitted and
if and only if
h(r, s) ~ 0 .
Since h is continuous in rand s, it
points
~here
~here
h = 0 is the
h < 0 and the set
follo~s
boundar~ bet~een
~here
that the set of
the set of points
h > 0, that is, this boundary di-
vides the (r, s) space into two regions, the region where h < 0
(that is, where decision d
il
is made) and the region where h
~
0
(that is, where decision d
is made). As it turns out, each of
iO
these regions is a connected set so the critical region (the retion where decision d
il
is made) is determined by these boundary
points together with the determination of the sign cf h at one other
point.
This boundary was determined by an iterative procedure with
the aid of the tables
k=10, 100, 1000.
LlL? and Ll§7,
and the solution was found for
These boundaries in the (r, s) space are given in
Figure l,(page 39).
The regions in the (r, s) space were then mapped
2
onto the (Yl' Y2) space for y = 0.5, 1, 3, and
(y2+ 1)/y2, ~ = 1 when y2=
00
00.
2
(Since ~ =
so for other values of y2 the regions
in the (Yl' Y2) space can be easily obtained from the regions for the
case y2=
and y2=
00.
00
Thus Figure 1 gives the boundaries for k=lO, 100, 1000
in the (Yl' Y2) space.)
39
Figure 1
Bayes decision procedure critical regions
(n
= 3, ~2o
known)
(third component problem)
s
4 g
It
:5
-
2
-
1
-
0
< 0
3
r
-"-_._._~---_.
g>O
3
-1
-3
= 10 -
k
-2 -
-
-4
_J~
-3
-2
-1
0
1
2
3
4
40
Viewed as above, the decision procedure is:
If the sample
x
)//2a, Ya= (x + x - 2x )/.[6a) is
means are such that (y = (X 2
l
l
l
2
3
a point in the region where g3 < 0, then make decision d ; that
31
is, place n, in the inferior group. Otherwise make decision d,O;
that is, leave n
in the superior group. However, it is easy to
3
see that an equivalent way of viewing this decision procedure is:
make decision d,l if
x,/j2a
l
< maX (Xl/.f2a, ~//2a)-t, where t is a
and t:. =
number determined by k, ..
I (xl/l2a)
- (x
2
//2a) I;
other-
wise make decision d • (In (Yl' Y2) space t is the distance
30
from the boundary where g,= 0, to the line x = xl i f xl~ X2 , or
3
to the line x,= X if X ~ X -). In the following this formula2
l
2
tion of the test procedure will be used since it is more useful
for practical purposes and is easier to generalize to cases where
n
> ,.
Table 1 (page 45 ) presents the values of t for this decision
procedure for k=lO, 100, 1000, y2= 0.5, 1, ,,~ and vcrious values
of 6 =
I (~/J2a)
- (~/j2a) I.
Thus we have the decision procedure for the third component
problem.
The decision procedure for the first component problem is
bbta.ined from that for
the third by interchanging xl and
X,
in
the decision procedure, that is, for the first component problem
make decision d
make decision d
6 =
ll
lO
if
xl /j2cr < max(x2 //2a,
i.,//2cr) - t, otherwise
' where t is now a function of k, y2 and
I (x2 //2a) - (x,//2a) I-
Similarly for the second component
x/
if X~/j2u
max (Xl!/20 ,
/20) - t ,
21
2
otherwise make decision d , where t is a function of k, Y and
problem, make decision d
20
i~
::;
I (x1 //2rr) - (x-./j2rr)
I.
)
i.s clear that thesl;.; compo.nent
It
decision procedures are compatible if and only if t is not less than
zero and it is easy to determine that t > 0 for k
not always non-negative for k < 3.
~
Thus to insure
th~
lity of the component decision procedures k ::; kl/k
o
Finally for k
~
3 while t is
compatibi-
must be > 3.
-
3 these component two-decision procedures consi-
dered simultaneously give the following decision procedure for the
seven-decision problem:
then place rt. in the inferior group,
1
otherwise leave rt. in the superior group
1
(i::; 1,2,3)
where t is determined by k, r 2 and
6 = max(
{xl /j2rr, x2 /j2rr) x3/j2rr)
-
x3/j2rr)
-
-min( (i/j2rr, X //2cr
2
j
Another way of stating this decision procedure is the following;
let x(l) ~ X(2) ~ x(3) be the ranked sampl~ means and
rr(l)' rt(2)' n(5) be the cOrTe8pand~ populations, then the superior
i.Sroup is
rt(l)
if (x(2/}2rr)
42
(2)
Jf(l) and 1(2)
if (X(2/12(J)
2:
(X(l/j2(J) - t 1
and (x(3)/j2rr) < (x(l)/j2(J)
and
(3)
-
\~,
Jf(l)' Jf(2) and Jf(3)
if
(x(3)/j2(J) 2:
(x(l)/j2(J) - t 2 ,
where t l is a function of k, y2 and (x(l)/)2(J) - (X(3)/j2(J), and
t 2 is a function of k, y2 and
(page 43)
ilb~strates
(x(l)/I2(J) -
(x(2)/j2(J).
Figure 2
the seven regions in (r, s) space correspond-
ing to the seven decisions.
For sample means such that (r, s) is
a point in region a. make decision d. (i ; 1, ... ,7).
~
Table 2 (page
~
49) presents the values of t for the decision
procedure for the case n ; 2 and various values of k and
y
2
2
and (Jo
known and unknown (v is the degrees of freedom for the estimate
2
so).
These t values were obtained by Duncan ~17 for the pairwise-
multiple comparisons problem as he treated it and are the appropriate values to use in the p-decision problem treated here for
n
= 2.
For n
=2
t is a function of k, y
2
and v but does not vary
with any function of the sample means (as in the case n
A comparison of t values for n
valent conditions for the case n
=2
=3
show~
= 3).
with the ones for equithat ten
= 3)
< ten
= 2).
This is of considerable interest in that it runs counter to a general tendency in previous approaches (to the problem of choosing
the largest mean) for such critical values to increase with n.
Figure 2
San~le
space regions for Bayes decision procedure
(n
= 3,
rr
2
known)
o
44
This increasing-with-n nature of the previous decision procedures
results from the method of Teducing the type-2-like error probability subject to the fixing of the type-l-like error probability in
the component problem6.
A more balanced emphasis placed on the two
types of errors in the Bayes approaah results in the decreasingwith-n nature of the critical values in the present decision procedure.
In Chapter IV it will be shown that this is a general
property of the Bayes decision procedure for all n.
45
Table 1
Bayes decision procedure t values, n = 3,
2
(r =00)
t:::.
0
.1
.2
·3
.4
e
·5
.6
.7
.8
·9
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3·0
3·5
5·0
e
k =
(J
2
0
knO\m
10
100
1000
.423
.470
.517
.558
.598
.632
.666
.696
.722
.746
.769
.806
.835
.857
.872
.882
.889
.895
.898
·900
.901
.901
.901
1.159
1.210
1.260
1.303
1.345
1.383
1.418
1.451
1.481
1.508
1.535
1.577
1.614
1.646
1.669
1.685
1.697
1.705
1.711
1.715
1.718
1.719
1.721
1.804
1.856
1.905
1.952
1.993
2.030
2.066
2.100
2.134
2.164
2.194
2.244
2.288
2·325
2·355
2.379
2.397
2.409
2.418
2.425
2.430
2.434
2.436
46
Table
1 (cont'd)
Bayes decision procedure t values, n
(:Y
b.
0
.1
.2
·3
.4
e
·5
.6
.7
.8
·9
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3·0
3·5
5·0
e
k
=
2
= 3, cr02
known
= 3)
10
100
1000
.423
.464
.503
.539
.575
.608
.638
.667
.695
.717
.738
.777
.807
.833
.853
.867
.877
.885
.891
.895
.898
.901
.901
1.159
1.203
1.245
1.283
1.321
1.357
1.389
1.421
1.450
1.475
1.498
1.5!l3
1.580
1.612
1.640
1.663
1.679
1.690
1.699
1.706
1. 711
1.718
1. 720
1.804
1.850
1.892
1.933
1.971
2.004
2.036
2.068
2.099
2.127
2.154
2.203
2.247
2.286
2.318
2.345
2.368
2.388
2.400
2.410
2.418
2.430
2.436
47
'table 1 (cont'd)
Bayes decision procedure t values, n = 3,
2
(J' = 1.)
6
0
.1
.2
·3
.4
e
·5
.6
.7
.8
.9
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3·5
5.0
k
=
(J'
2 known
0
10
100
1000
.423
.456
.489
·521
.552
.579
.606
.630
.654
.677
.697
.734
.766
.794
.817
.837
.853
.865
.873
.880
.886
.896
.901
1.159
1.195
1.230
1.263
1.295
1.324
1.353
1.380
1.405
1.430
1.453
1.493
1.552
1.563
1.591
1.625
1.640
1.658
1.673
1.684
1.692
1. 707
1. 720
1.804
1.842
1.878
1.912
1.945
1.974
2.000
2.026
2.052
2.078
2.103
2.149
2.190
2.228
2.261
2.292
2.317
2.340
2.360
2.377
2·391
2.411
2.435
48
~able
Bayes decision
proced~re
(7
6
0
.1
.2
e
·3
.4
.5
.6
.7
.8
·9
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3·0
3.5
5.0
e
k
=
2
1 (cont'Ci)
t values, n
= 3, (J"02
known
= ·5)
10
100
1000
.423
.450
.477
·503
.529
.553
.575
·597
.618
.638
.658
.695
.724
.751
.777
.798
.817
.833
.847
.858
.867
.882
.900
1.159
1.188
1.217
1.245
1.272
1.297
1.321
1.345
1.368
1.389
1.410
1.450
1.483
1.514
1.543
1.568
1.590
1.612
1.632
1.648
1.663
1.687
1.717
1.804
1.834
1.864
1.892
1.920
1.947
1.971
1.994
2.015
2.036
2.057
2.099
2.137
2.172
2.203
2.233
2.260
2.286
2.308
2.327
2.345
2.381
2.427
Table
2
*
Bayes decision procedure t values, n
=2
,
·l
Log k
,1..~2.0
v
0.0
0·5
1.0
1
0.0
.375
.807
1.353
2
0.0
.413
.860
4
0.0
.434
6
0.0
14
2·5
3.0
3·5
2.102
3.160
4.685
6.854
1.379
2.012
2.814
3.851
5.208
.884
1.367
1.900
2.502
3·197
4.010
.443
.891
1.356
1.848
2.374
2.948
3.580
0.0
.451
.898
1.340
1.779
2.217
2.654
3.099
co
0.0
.457
.902
1.326
1. 721
2.091
2.436
2.759
1
0.0
.444 1.053
2·503
co
00
00
2
0.0
.484 1.060
1.926
4.077
00
00
(l)
4
0.0
.506 1.056
1. 718
2.623
4.178
9·595
00
6
0.0
.515 1.053
1.653
2.370
3.308
4.732
7.706
14
0.0
·522 1.047
1.582
2.136
2.724
3.360
4.074
(l)
0.0
.528 1.041
1.531
1.987
2.414
2.813
3.186
(l)
3
*Reprinted
from ~17.
co
50
Table 2 (cont'd)
Bayes decision procedure t values} n = 2
y
2
Log k
0·5
1.0
0.0
.572
1·930
0.0
.610
1.532 8.741
4
0.0
.629
1.395 2.648
8·592
6
0.0
.637
1.353 2.303
3.980 13·625
14
0.0
.642
1.308 2.030
2.859
3·891
5.326
7.818
00
0.0
.646
1.275 1.875
2.433
2.957
3·445
3·902
1
0.0
.767
2
0.0
.785
2.292
4
0.0
.791
6
0.0
14
00
11
2
1
e
·5
1.-5- 2.0
0.0
v
00
2.5
3.0
3·5
co
00
00
00
co
00
00
co
00
co
00
00
00
00
00
00
00
00
00
00
en
00
00
1.963 9.243
00
00
00
00
.794
1.800 3·777
00
00
00
00
0.0
.792
1.653 2.693
4.162
6.670
00
00
0.0
.792
1.562 2.296
2.980
3.622
00
4.219
4.779
CHAPTER IV
SOME DERIVED DECISION PROCEDURES AND THEIR PROPERTIES
4.1
Progressive method for generating a class of decision proce-
dures for the p-decision problem.
The folloWing notation will be used:
L:
p-decision problem of choosing the largest mean, as
presented in Chapter II,
L : the i
i
th
component two-decision problem used to generate
L,
D : a decision procedure for L in which the region for
i
i
making d
is denoted by Ai' (Using the language of
il
hypothesis testing A. will be called the critical
~
region. )
D:
a decision procedure for L,
D.(B):
a Bayes decision procedure for L.,
~
D(B):
~
a Bayes decision procedure for L.
As shown in Chapter II, a compatible set of Bayes decision
procedures, D.(B) (i
~
problems L. (i
~
= 1,2, ••. ,n),
= l, ..• ,n)
for the component two-decision
generates a Bayes decision procedure,
D(B), for the p-decision problem L.
Moreover, any compatible set
f
of decision procedures, D. (i
~
two-decision problems, L. (i
~
= l, ..• ,n),
= l, •.. ,n),
for the component
generates a decision
procedure,
p, for the p.deciilQn problem, 4. Also/ due to the
symmetrical way in which the means and loss
funetiou~
havs been
treated in these component problems, it follows that a decision
procedure for any component problem, with only changes in nota·
tion, will yield a decision procedure for each one of the other
component problems.
That is, from a decision procedure for one
component two· decision problem we have decision procedures for
all n of the component two-decision problems and, if these twodecision procedures are compatible, they generate a decision procedure for the p-decision problem.
Therefore, to obtain a class
of decision procedures for the p-decision problem with a symmetrical treatment of the means it will be sufficient to obtain a class
of decision procedures for a component two-decision problem.
Consider the i
th
component two-decision problem, L , with
i
the two decisions:
d iO : I-l
€
O)iO = .f\.
d il : I-l
€
O)il = [~j I-l i < I-lmax }
Remove one population other than
1C.,
say
1
te
j
(j
# i),
and con-
sider the decision problem L~, which is a two-decision problem
1
where the mean of
te.
1
populations except
is compared with the means of all the other
That is, for this two-decision problem
te .•
J
the two decisions are
J
d iO : I-l
€
(DiO
dil: I-l
€
(.Dil =
j
= Sl
{~;
I-l i
:: max(l-ll,···,l-lj_l,l-lj+ly···lJ.n)}
53
Let A~ be the critical region fo!' a decision J:ir()Cec.url~
1
From D~J_ decision procedures
this two-decision problem.
(k
= l, •.. ,n;
k
~
ob~ainable
i) are easily
metrical treatment of the
D':
1
for
D~1
due to the sym-
Then for the two-decision prob-
~'s.
lem D. we can generate the following decision procedure:
1
make decision d
and make decision d
(j
= l, ... ,i-l,
that is, the region Ai
il
iO
if d
j
il
d~o
if
is made for any j,
is made for all j,
i+l, ... ,n),
wh~re
decision d
il
is made is the union
n
A.1 =
U A~1
.i=l
j#
and the region Y-A where decision d
is made is the intersection
i
iO
n
Y-A.1
= (J
j=l
j#
(This is the union-intersection method of test construction as
proposed by Roy
[2g7.)
Thus a one-versus-(n-2) procedure (that is, a decision procedure for a two-decision problem in which the mean of one population is compared with the means of (n-2) other populations) can be
used to generate a one-versus-(n-l) procedure.
Carrying this one
step further, a one-versus-(n-3) procedure can be used to generate
a one-versus-(n-2) procedure.
Combining these results, a one-versus-
(n-3) procedure can be used to generate a one-versus-(n-l) procedure.
Hence, it is easy to see that a one-versus-(n-m-l) procedure
(1 ~ m ~ n-2) can be used to generate a one-versus-(n-l)
procedure (which is a decision procedure for a component twodecision problem).
In the above terminology, the Bayes decision procedure
obtained in Chapter III for a component problem with three
populations is a one-versus-two procedure.
Also the Bayes de-
cision procedure obtained by Duncan ~17 is a one-versus-one procedure.
Hence for each of these known decision procedures a de-
cision procedure for the p-decision problem can be obtained as
above.
Also, as the necessary integration is done and the Bayes
decision procedures are found for cases in which there are more
than three popUlations, each can be used to generate a decision
procedure for the general p-decision problem.
In section 4.2 some important and interesting relations
between the decision procedures generated as above from Bayes
decision procedures for subcomponent problems will be given.
These relations will in turn show the significant-differencedecreasing-with-n nature of the Bayes decision procedures.
In
section 4.3 special attention will be given to the decision procedure for the p-decision problem obtained as above from the Bayes
one-versus-one decision procedure.
4.2
Some properties of the generated decision procedures.
The folloWing notation will be used in the remainder of
this chapter:
55
a decision procedure for the component
t~o-decision
problem L.,
~
D.(B):
a Bayes decision procedure for the same problem,
~
.lm
DJl'''''
:
i
a one-versus-(n-m-1) procedure for the
decision problem where
~i
t~o-
is compared with the means
of all the other populations except
1t. , ••• ,
1t
J1
jm
(j1, ••• ,jm are some m of the integers 1, ... ,i-1,
i+1, •.• ,n)
j1' ..• , jm
D.
(B):
~
a Bayes decision procedure for the same
problem,
D.(B
~
n-m
):
L
i
a decision procedure for the component problem
obtained by the union-intersection method of test
construction from the Bayes decision procedure
j1,···,j
D.
m(B) (1 < m < n-2),
- -
~
D
D(B):
a decision procedure for the p-decision problem L,
a Bayes decision procedure for the p-decision prob1em L.
D(B
):
n-m
the decision procedure for the p-decision
problem L generated by the decision procedure
D.(B
) for the component ·two-decision problem
~
n-m
The following definitions are made:
(1)
The decision procedure D. (for the component two-decision
~
problem L ) is said to be conservative with respect to a second
i
56
decision procedure Di
(for the Bame problem) i f the critical
region Ai for D is contained in the critical region Ai for Di.
i
(That is, AiCAi and Ai # Ai')
(2) The decision procedure D (for the p-decision problem L) is
said to be conservative with respect to a second decision procedure D' (for the same problem), where D and D' are the decision
procedures for L generated by the decision procedures D and
i
Di, if D is conservative with respect to Di for i
i
= l, ••• ,n.
If it is recalled that for a point in the critical region
the decision to place
1(.•
J.
in the inferior group is to be made
while for points outside the critical region the decision to
leave
1f i
in the superior group is to be made then if D is coni
servative with respect to Di it is seen that D leaves
i
the superior group more often than
D~.
J.
1(
i in
It follows that if D is
conservative with respect to D' then for a given set of observations D makes a decision which never leaves fewer populations
in the superior group than D'(moreover, the superior group obtained with D contains the superior group obtained with D' ).
It
follows from definition 1 that if D is conservative with respect
i
to D~]. and D~]. is conservative with respect to D~'
than D.]. is conJ.
servative with respect to Di' (since if AiCAi and AiCAi' then
AiCAi').
From this and definition 2 it follows that if D is
conservative with respect to Dr and D' is conservative with
respect to D"
then D 1s conservative with respect to D".
5'7
The important result that Di(B _ ) is conservative with
n l
respect to Di(B) is given by the following theorem.
Theorem
Let Di(B) be a Bayes decision procedure for the i
th
com-
ponent two-decision problem with the two-decisions
diO: IJ.
~
<.J.)iO = S'L
d
~
')
<.J.)il = {~) IJ..1 < IJ.max)
il
: IJ.
and loss functions and prior distributions as given in Chapter II.
Let D.(B
1) be a decision procedure for the same decision system
nobtained by the union-intersection method of test construction
1
from the Bayes decision procedure for the two-decision problem
with the two decisions
and loss functions and prior distributions as given in Chapter II.
Then the decision procedure D.(B
1
n-
1) is conservative with respect
to the decision procedure D.(B).
1
Before proving the theorem we first illustrate it with an
example.
For n = 3 and
2
known D (B) is as given in Chapter'III
3
either by Figure 1 or Table 1. The critical region A (in (Yl' Y2)
3
space) for D (B) is given in Figure 3 for a particular k and 1 2
3
(which need not concern us here). D (B) is a Bayes decision pro0"0
3
cedure for the problem where 1J. is compared with
3
1J.
1
and 1J.
2
.
For
58
a Bayes decision procedure for
with
th~
problem where
~3
is compared
~l
we have the decision procedure D (B) from Table 2.
2
Figure 4 gives the critical region for this decision procedure
with the same k and r
2
as used for D (B) in Figure 3.
3
Similarly
Figure 5 gives the critical region for D (B) for the problem
2
where ~3 is compared with ~2'
The critical region of D (B _ )
3 3 1
is the union of the critical regions in Figures 4 and 5 and is
given in Figure 6 together with the critical region of D (B).
3
From this figure it appears that the critical region of D (B _ )
3 3
is contained in the critical region of D (B).
1
The theorem states
3
that this is indeed the case and holds for any n.
We now return to the proof of the theorem.
Proof:
The result of the theorem will follow if it can be shown
that the critical region, Ai(B _ ), for the decision procedure
n l
D (B _l ) is contained in the critical region, Ai,(B), for the dei n
cision procedure D.(B).
~
With no loss of generality let i
Since
if
for J
=
2, ••• ,n.
= 1.
59
The critical regions
A (B), A2 (B), A1 (B), A (B2)
3
3
3
3
"-~ '~iI
i
-'-.i..
Figure 5
I I
A~(B)
-+--~--_.-+--_._----_.
J
Figure 4
Figure 3
I
f
~
III
A~(B)
Figure 6
60
But due to the symmetry of our treatment of the
~IS
and the loss
functions it is easy to see that if
then
Therefore it is sufficient to prove that
A~(B) C Al (B) .
But
I
€
Al(B) if and only if
i f gl(Yl' .. "Y - )
n l
Bl(l) < 0 and I
€
A~(B) if and only
< 0 (where the function gl is as given in
section 2. ·5) •
Therefore
if
gl(Yl ' ... , Yn ) < gl(Yl ' ••. } Yn - l ) •
Since if gl(Yl ' •.• , Yn ) < gl(Yl ' .•. , Yn - l )
then
gl(Yl ' .•• , Yn - l ) < 0 implies gl(Yl ' .•• , Yn ) < 0
so
I
that is,
€
A~(B) implies I
€
Al(B),
61
From section 2.5
n
gl(Yl'''''Yn ) :; -(ko/a-) ~
0
J
(I-l/~) - 1J.1(~»f(:t; S:)~(~)dQ
j-21l~ {~)lJ.j> max {lJ. j })
+(k/rr)
n
j~2J (~l(~) - ~j(~))f(D~a(~)d~
~ n (£;lll>IJ.j>max {lJ.l,Il J }}
and
In the expression for gl(Yl""'Y ) separate the (n-l) terms of
n
the first sum(: (ko/a-)r;. •• ) into the sum of the first (n-2) terms
and the term with J :; n.
Next divide the region of integration
for this last term (where
j :;
n) into the (n-l) regions,
and then group the (n-l) integrals obtained from this sUbdivision
of the region of integration into two parts, the first with 1
and the second consisting of the remainder.
~
1
Separate the (n~l)
terms of the second sum «kl/a-)n •.• ) into the sum of the first (h-2)
terms and the term with j
e
==
h, then divide the region of integra-
tion for this last term (where j "" n) into the (n-2) regions,
JL*
Q
n {£;
III > IJ.n > lJ. i > max {J...L1 , jJ.i' Ilnl}
,
(i
= 2, •.• ,n-l),
62
to
obtain
-(kO/crU (~n(~)-~l(£)) f(X; £) ~(£)d~
. n.=nt~;~n>
~1>
max
{~l'~n}\
In the expression for b1(Y1,""Y - ) divide the region of inten 1
eSration for the (n-2) t~rms in the first sum«-k /u)r. •.. ). inte the
Q
two regions
and
divid~
the region of integration for the (n-1) terms in the second
sum «kl/cr)~ ... ) into the three ~gions
and
to obtain
64
(kola)
J(Iln(~)-1l1(~»f(lj ~) s(~)d~
n= n
{~';Iln>
Ill> max {1l1 ,lln }}
In this diffGrence the first term is positive since the integrand
is positive over the region of integration.
are each positive for the same reason.
(n-2) terms.
The next (n-2) terms
Similarly for the next
Finally, the last (n-2) terms are each positive for
the same reason.
Therefore we have
that is,
which implies
which implies that D (B
1
n-
1) is conservative with respect to
Dl(B») and since there was no loss of generality in considering the first component problem we have that Di(B _ ) is
n 1
65
conservative with respect to D (B).
1
(It should be noted that the above argument is not limited
to cases with the loss functions and probability density functions
f(r;
~)
and
s(~)
of Chapter II but applies to other much more
general situations.
Instead of the loss being proportional to
the difference between two means it could be any non-decreasing
function of the difference.
f(l;
~) and
s(£)
Also the probability density functions
can be any probability density functions with
only the restriction that the Bayes decision procedure is determined by the functions g(l)'
* does not contain the paraSince SlQ
meter points where any of the
~'s
are equal, these points must have
E-measure zero. (However the above argument can be modified to
overcome this difficulty.»
In terms of the t values used in the decision procedures
the theorem states that for a given set of sample means the t
value used with the decision procedure D.(B
~
n-
1)' say t LD·(Bn-1)7,
~
is greater than the t value used with the decision procedure
From the theorem and the transitive property of conservativeness the following corollary results.
Corollary 1
The decision procedure D.(B
) is conservative with
~
n-m
respect to the decision procedure Di(B). In particular, D (B )
i 2
is conservative with respect to
Di(B), (1
~ m ~ n-2).
66
In tarms of
th~
t valu0s usad in
thes~
decision procedures this
corollary states that
There are two interpretations of tf5.{B
)7.
LJJ~ n-m-
)7
is that tf5.{B
LJJ~ n-m-
The first
is the t value for a decision procedure for
the two-decision problem L. where ~. is compared with (n-l) other
~
means.
~
(This decision procedure is obtained by the union-inter-
section method of test
co~ction
from the Bayes decision proce-
dure where ~. is compared with (n-m-l) other means.)
~
The second
)7
interpretation is that t/Di{B
is the t value used in the
n-m-Bayes decision procedure for a two-decision component problem Li
when there are only (n-m) populations in the problem.
With this
second interpretation we have the following corollary.
Corollary 2
When the number of populations included in the problem is
increased the t value for the Bayes decision procedure decreases.
This is the generalization of the result noted in Chapter III
with two and three populations.
As noted there this decreasing-
with-n behavior of the Bayes decision procedure t values is of
considerable interest since it is opposite to the behavior of decision procedures proposed for the problem of choosing the largest
mean by previous workers using fixed-type-l-like-error-probability
approaches.
67
4.5
Th~ decision procedur~s D.(B~) and D(B r
].
<::'
c;
).
In this section we will obtain the decision
proc~dure
D.(B ) for the two-decision problem L. obtained from the Bayes
]. 2
1
decision procedure for
of
11: j
1C •
].
two-decision problem where the mean
th~
is compared with the mean of one other population (say
where J
I:
1).
For D (B ) (1
i 2
= 1, ••• ,n)
the condition of
compatibility is determined and then the decision procedure
D(B ) for the p-decision problem L is obtained.
2
For the two-decision problem where the mean of
compared with the mean of
A
Bay~s
1C
•
J
11:
i is
the two decisions are
diO ( j ): Il
€
(1)iO (j) = SL
dil(j): Il
€
(J)il(j)
= {~;
Il i < Il j }
decision procedure for this problem with assumptions
which imply the assumptions listed in section 2.4 was found by
Duncan ~17.
Thus the decision procedure found by Duncan is
the Bayes decision procedure for n = 2 for a component problem
that generates the p-decision problem L.
This decision proce-
dure can be written as
make decision diO(j) i f (x.//2s) > (x//2s)-t
J.
and make decision dil(j) i f (x//2s) < (i../j2s)-t)
J
where t is a function of k, v and
2
r.
Therefore, using the
union-intersection method of test construction, the decision pro:.:edure fcr the two-decision pro11atn \.here the illean of
1C.
].
is compared
68
with th<;; means of two other
~opulations,
say
and
~.
Jl
~.
J2
is
makG decision d (jl,j2)if (x /j2s) > (x. /j2s)-t
iO
i
J
l
and if (x./j2s) > (x. /j2s)-t
~
J
2
or if (x./J2s) <
~
(x.J /J2s)-t
2
that is,
Hence it is easy to see that the decision procedure D (B ) for the
i 2
component two-decition problem L , generated from the above decision
i
procedure as outlined in section 4.1, is
make decision d
iO
if (x.//2s) >
~
-
and make decision d~l if (x.//2s) <
...
~
These component decision procedures (i
.
max
('X..//2s)-t
J=l, .•• ,n
max
J
(x./J2s)-t.
. 1 , .•• ,n J
J=
= l, ... ,n)
t is not less than zero which is true for k
=
are compatible if
kl/k
> 1. Thus for
o -
~
1 the decision procedure D(B ) for the p-decision problem
2
generated by these n compatible two-decision procedures is:
k
~
(1)' ~ (2)"'" '~tn) be the corresponding populations
and
l~t
j (an integer between 1 and n) be such that
and (X(j+l)//2s) < (X(l)/)2s)-t;
then the superior group is
1C(1)' ..• ,1C(j)
and the inferior group is
(where t is a function of k, v and
1
2
but independent
of the sample means and is given in Table 2).
For reasons explained below we will refer to this decision procedure as a conservative-near-Bayes decision procedure.
A comparison of this conservative-near-Bayes decision procedure with the Bayes decision procedure for the case n
=3
as
found in Chapter 3 shows that it is somewhat easier to apply since
here t does not vary with any function of the sample means as it
does in the Bayes decision procedure.
It can also be seen from a
comparison of the t values from Tables 1 and 2 that for the case
n
=3
if the ranked sample means, x(l)
~
x(2)
~
x(3)' are such
that (X(l)- X(3»/j2~ is large then the Bayes t value (Table 1)
and the conservative-near-Bayes decision procedure t value (Table
2) are approximately the same.
This is quite reasonable to ex-
pect since if (X(l)- x(3»/j2~ is large then the problem reduces
essentially to a problem with onq the two means X(l) and X(2) .
70
For this reason, among the
cl~ss
of decision
procedur~s
for
the p-decision problem which have t values independent of the
sample means, the decision procedure D(B ) is the 'closest'
2
conservative approximation to D(B) for all n.
(vfuere 'closest'
is used in the sense that tLD(B217-tLD(B17 is smaller than for
any other conservative decision procedure in the class of decision procedures with t independent of the sample means.)
This gives justification to the use of the term conservativenear-Bayes decision procedure for D(B ).
2
4.4 Comparison of risk functions of D(B) and D(B21.
Let
Rl(~)
be the risk function for the conservative-near-
Bayes decision procedure D(B ) and R2(~) be the risk function
2
for the Bayes decision procedure D(B); then it is of interest
to find the ratio R /R as another measure of the closeness of
l 2
D(B ) to D(B). This ratio has been evaluated for various values
2
of Q in the case n = 3 and ~2 known with k = 10 and y2 = 1.
o
As pointed out in section 2.2c, since the loss functions
for the restricted product problem are the sum of the loss
functions for the component problems and since expectation is
a linear operator, the expected loss, or risk, for the restricted
product problem is the sum of the risks for the component problems.
That is,
R.(Q)
J -
=
3
~ R.. (Q)
where Ri.(Q) is the risk function for the i
J -
(j
.
J.JJ.=l
th
= 1, 2)
. component decision
71
procedure which generates the decision prOcedure n(B ) (conser2
vative-near-Bayes decision procedure) for j
=1
and generates
the decision procedure D(B) (Bayes decision procedure) for j
For the i
th
= 2.
component problem
1
R.. (Q) = L: L(d. ; Q) P.(d. ; Q);
J.J -
where L(d
~)
;
ik
J. k
k=O
-
J. k
J
-
is the loss incurred by making decision d
ik
when Q is the parameter point and is as defined in section 2.2c,
and
P/ dil ;
~)
=
J <.~~) ~)dl'
f
A
ij
where A.. is the critical region for the i
J.J
using the decision procedure D (B ) for
i 2
= 2.
procedure D.(B) for j
J.
making the decision d
Di (B2 ) for j
=1
ik
(P.(d. ;
J
J. k
component problem
=1
and the decision
is the probability of
for the two different decision procedures,
and Di(B) for j
Using the tables
Q)
-
j
th
LI£7
= 2.)
and the method described therein for
determining bivariate normal probabilities over polygons
Pl(d ik ;
£)
of Ail (i
(k
= 0,
(k
= 1,
= 0,
1) can be evaluated exactly since the boundaries
To evaluate P (d ; ~)
2 ik
( i = 1, 2, 3) can be approximated
2, 3) are straight lines.
1) the boundaries of A
i2
by a series of straight line segments and the above tables and
72
T
method used.
tt is easy to show that
T2
where eATl and eA are the images of
~
T
R.(9)
= R.(9
1) = R.(Q 2)
J J J
under 1200 and 2400 rota-
tions of the (9 , 9 ) axes. Thus RJ(~) needs only to be evalu2
1
0
ated over a 120 sector of the ~ plane to determine it everywhere.
0
Also it can be shown that for the proper choice of this 120
sector there is a symmetry under a reflection in the bisector of
the sector.
Hence R.(Q) needs only to be evaluated over a 60
0
J -
sector.
This evaluation has been done for various points in the
0
sector 9 2 ~ G //3 ~ 0 and the resulting values of Rl(~)/R2(~)
l
0
are as given in Figure 7. From the 60 sector in the figure a
60
reflection in the line 9
sector 91 ~ 0, -~1/13 ~
= 9 1//3 will give values for the 600
9 ~ 9 //3 and this sector together with
2
1
2
0
the sector in the figure under rotations of 120
give values for the remainder of the 9-plane.
and 240
0
will
73
E!gure 7
Ratio of risk for conservative-near-Bayes
decision procedure to risk for Bayes decision
= 3, ~2o
procedure (n
known)
Q
2
2..G7"2
I
I
/
e
/
*1.184
........
"
1
"
I
....
"
/
"
\
\
/
/
l.5g8
/
,.'t.1.239
.....
/1. 013
,.
....
I
)Ie"
'600~
A,. 942
/'
\.
-:841
/'
13
,
1= ·5
Q
e
/
/'
,/
I"
J
/
"-
/
./
/
"
~·.3t3 ~686
I
./
"-
/
1/ ,.
,/
"-
/
./
I
1.0
I
2.0
Q
1
CHAPTER V
THE EFFECT OF DEPARTURES FROr-1 NORMALITY
IN THE PRIOR DISTRIBUTION
5.1
Introduction.
In this chapter results are given of a study made of the effect
on the Bayes decision procedure due to departures from normality in
the prior distribution of the ~'s for the case n = 2 and rr~ known.
Since n = 2 these results apply not only to the Bayes decision procedure for the p-decision problem for p = 3 (= 22_ 1) but also to
the ~a1rwise multiple comparison problem as treated by Duncan ~17
for all n and to what we have called the conservative-near-Bayes
.
decision procedure (D(B » for all n.
2
5.2
Choice of a family of prior distributions.
Let
T)=E(~.)
J.
var(~.)
J.
= E(~.J.
T)
2
~= E(~-T)3/LVar(~173/2
and
E(~-T)4/LVar(~172
04=
, (skewness coefficient),
,(kurtosis coefficient).
For the study of this chapter it was desired to choose a family
of prior distributions of the
values of 03 and
~'s
which would give wide ranges of
(since they are often used as measures of de4
parture from normality). Also it was desired that this prior disQ
tribution should not differ much from the previous normal prior
75
distribution if
°3 and 0;4 did
not differ much from
a normal distribution (where ~= 0,
°4= 3).
°3 and °4 for
With such a prior
distribution the study is essentially reduced to a study of the
variation of the Bayes decision procedure with the variation of
°4 of
the new prior distribution. Since· the moments,
~
and
and~speciully
only the first four moments, do not always completely determine a
d.istribution the above reduction in the study means that it is not
complete since it will not be made with all possible prior distributions.
However the study is at least a first approximation to a
complete one.
From a practical point of view with such a prior dis-
tribution an experimenter who is unwilling to assume that the prior
distribution is normal but who is willing to assume that the prior
distribution is determined close enough for his purposes by the
first four moments could use the Bayes decision procedure with this
new prior distribution.
A relatively simple distribution with the desired properties
is the fOllowing one obtained as a mixture of two normal distributions ,
76
However this distribution is hot completely satisfactory for our
purposes since the first four moments do not completely specify
all the parameters.
To overcome this the condition
pb 2 = ( l-p ) b 2
1
2
= 0 imply
3
symmetry of the distribution.) Therefore the prior distribution
was imposed.
(This condition was suggested by having
0:
of the population means used is given by equation 5.2.1 with the
additional restriction that
(For this distribution,
0:
=0
3
symmetrical about the mean.)
implies that the distribution is
With the proper choice of parameters in this distribution the
variance can be set equal to the value used previously and a complete range of values of
and
0:
3
4 obtained.
0:
Also when
0:
= 0,
3
4= 3 the above distribution reduces to a normal distribution as
0:
desired.
The choice of this distribution is in keeping with some
previous work of Karl Pearson's
1217 in which a mixture
of two nor-
mal distributions was used to fit some non-normal distributions.
Finally with this choice of prior distribution the necessary numerical integration can be carried out with the aid of existing
tables
/J-il.
•
5.3 Parameter values used.
For this study
Table 3.
~
and
4 values were used as indicated in
0:
77
Table 3
0:3' 0:4 values used
0:
3
0
1
2
X
X
3
X
""
0:4
3
*
*
5
X
6
X
X
X
*
10
e
X
11
X
X
X
X
18
v
X
X
X
"
('X' indicates 0: , 0:
4 combination used.)
3
Those (0: , 0: ) combinations with an asterisk are for the limiting
3
4
form of the prior distribution where b
l
= b 2 = O.
This limiting
form is
\j.r(lJ. )
i
=
IJ. i
= )'cr( c +e)
= (l-p)
lJ.
= ),cr{c-e)
=
elsewhere
p
0
i
(i = 1, 2),
that is, the distribution where IJ.. takes the value )'cr(c+e) with
l.
probability p, the value 7cr{c-e) with probability (l-p), and other
values with probability zero .
•
It is easy to see for the distribution in equation 5.2.1 that
78
var(lJ.)
~ 2 (2
2 I' 2cr 2
= LPb
+ l-P)u +4p(1-p)e_
7
1
2
E(IJ._~)3 = Lbp(1-p)e(bi-b;)+8p(1-P)(1-2P)e~713cr3
and
E(IJ._~)4 = {3LPbi+(1-P)b~_7+24p(1-p)e2Ltl-p)bi+Pb~_7
_7e 4 } I' 4cr4
+16p(1-P)L1-3P+3p2
•
Thus imposing the condition
pb
and setting
2
1
= (1_p)b 22
var(lJ.)
= r 2cr2
as before it readily follows that
and
03
= (1-2p)L5e-4p(1-p)e 3_7
04
= 3L4P(1-P17- 1 +
6(2p_l)2e 2. 4p(1-p)( 12p2. 12P+5)e
4
.
For the limiting (b = b = 0) distribution
2
l
var(lJ.)
= 4p(1_p)e212cr2
E(IJ._~)3 = (1-2P)8p(1-p)(elcr)3
and
E(IJ._~)4 = 16p(1-P)Ll-3P(1-p17(elcr)4
r.::.
--1/2
thus 0: = ( 1-2p )LP(l-pll I
3
l
and 0:4 = [P(1-p17- - 3 = ~ + 1 .
From the above formulas the parameter values corresponding
to the
0)'
0: values given in Table 3 Were found and are as given
4
in Table 4 (page 79). (For our purposes c need not be specified.)
The distributions for these parameter values (With c
•
sketched in Figure 8 (page 80).
= 0)
are
79
Table 4
Parameter values used
in non-normal prior distributions
e
(CXy cx4)
(b 1> 0, b2> 0)
(0, 2)
e
.6180
b1
b2
.7862
.7862
p
·5
1.
(0, 3)
0
·5
1.
(0; 6)
0
.1464
1.8478
.7654
(0,11)
0
.0736
2.6064
.7347
(0,18)
0
.0436
3.3878
.7230
(1, 3)
.9899
.2727
.6387
·3912
(1, 6)
.5270
.1666
1.5932
.7123
(1,11)
.4019
.0787
2.4607
.7192
(1,18)
.3695
.0453
3.2823
.7151
(2, 6)
1.3217
.1452
.6758
.2785
(2,11)
·9112
.0955
1.9324
.6279
(2,18)
.7728
.0513
2.9360
.6825
(3,11)
1.7323
.0837
.6904
.2086
(3,18)
1.3155
.0615
2.2093
.5656
(1, 2)
1.1180
.2764
0
0
(2, 5)
1.4142
.1464
0
0
(3,10)
1. 8028
.0840
0
0
(b 1= b2= 0)
e
80
Figure 8
Non-normal prior distributions
0:
3
0:
4
=1
=6
81
Figure 8 (cont1d)
Non-normal prior distributions
a
3
= 2
04 = 6
a =3
3
Q4 = 11
82
Figure 8 (cont1d)
Non-normal prior distributions
83
From this it can be seen that qUite large devihtions from the
normal distribution have been obtained with the famEy of distributions employed.
5.4 Bayes decision procedure.
As described in section 2.5 the Bayes decision procedure
for, say, the first (of the n
make decision d lO if gl
and
make decision d
ll
~
= 2)
component problems is:
°
if gl < 0,
where
=
r
-(ko/crU .@2(~)-~1(9)7f(l, ~H(~)d~
~2>~1
('
+(kl/crV
ffil (~)-~2(Wf(~~)
~)~(~)d~
.
~1>IJ.2
Also
and
Where
•
~(~)
is the prior distribution of the standarized difference
of the two population means, while W(~) is the prior distribution
of one population mean.
84
(1)
b l > 0, b2 >
° case.
g(g) can be found from
W(~.)
with the use of characteristic
1
functions.
E{exp( itg17 = EFXP(it~1/j2cr17 Elexp(.it~2//2cr17
= {p expjJ.(c+e)1t/!2_7 exp(-bi12t2/4)
+(1-p)expjJ.(c.e)1t/!2_7 exp(_b~12t2/4)}
{p
expEi(c+e )yt//2_7 exp( -bi12t2/4)
+(1-p)expEi(c-e)1t//2_7 exp(_b~y2t2/4)}
222
222
= P2 exp(-bl1
t /2)+(1-p)exp(-b 2Y t /2)
+p(l-p)expjJ. j2e1!:.7 expE(bi+b;)y2t2/1£!
+p(l-p )expEi !2e1!:.7 expf'(bi +b~)y2t2/~l.
From which it follows that
= p2(2n)-1/2(b 1)-1 exp;:(2b 212 )-1 g2 7
1
-
1
-
+(1_p)2( 2rry l/2(b y)-l expE(2b 212 ).lg2
2
7
2-
E(
2
+p ( l-p )( 2n ) -1/2~(
v 2 b 2l +b2)-1/2
1-1exp - b 2l +b 2)-1
Y -2 (g-e1)_
7
2
2
+P(1-p)(2rr)-1/2j2(bi+b~)-1/21-lexpE(bi+b;)-11-2(g+e1)=7.
Therefore
•
85
where
2
a
all
=p
a
= a 14 = p(l-p),
a
13
21
j
= bll' ,
a 23 = a 24 =
a
31
12
=
(l-p)
2
,
a 22 = b 21"
vlbi+b~ 1'//2 ,
= a 32 = 0
,
a33 = /2e-, ,
a34 = -/2e-, .
By completing the square of Q in the exponent of fey; Q)s(Q) we
have
fey; Q)s(Q)
=
4
E alo(2n)-1/2c~1 expL:(y-a o)2/2C 2 7(2n)-1/2s~1 expL:(Q-mo)2/2s2i 7
i3~
i=l ~
~
~
~-
where
C
i
= (1+a2
mi
= (a21y+a3i)/(1+a2i)
s.~
= a2.1.~1+a~.)
~ I'
c;~
Let d(
~ x; a,b)
then
gl(y)
2i
)-1/2
2
= ( 2n )-~2 b -1
2
(i
2, 3, 4).
E
2exp -(x-a) 2/2b_1
o
= (ko/~) j<12~Q) f(yjQ)~(Q)dQ
-00
= 1,
co
+
(kl/rr~(/2~Q)f(Y;Q)s(Q)dQ
0
•
m. ,
1.
So
~
)dQ .
86
then
4
K.(Y) =
-i
12k0
L
-m,/s.
0.
1=1
' ¢(y; a_.,c.)
1~
)1 1
¢(z.)dz.
J'~s~z.~m.)
1 1
1
1
1
-00
Jm (
00
0.
.,c. )
31 1
s z. ffil.) ¢( z. ) d Zi
111
-mi /s l
4
=
12k0'1
L 0. '
1~
1=
¢(y; a .,c.) s.
~
3~ 1
J
z.1 ¢(z.~ )dZ
-00
00
s.Jz.
1
But
L
0.
,
1=1 1 ~
J,;a
3
-mi /s l
¢( z)dz = -¢(a),
J-¢(z)
dz
=1
-
-00
00
J
Z ¢(z)dz = ¢(a)
-a
00
and
J
¢(z)dz
-a
-mi /s l
¢(z.)dz.
1
1
¢ (y j a 1·,c.)
m.J¢(z.)dZ
..
~
1
1
1
-00
•
~
00
4
+ fi~
1
- i /s l
= ~(a)
~(a)
,
i
87
where
a
and
J
=
!(a)
(21t) -1/2 exp( _z2 /2}dz.
-00
It readily follows from the above that
= j2
gl(y}
= j2
4
~ a · ¢(y; a , c.}s. ¢(m./s.)
3i
0 '1=
1 l1
1
1
1
1
(k1-k)
4
(kl-ko ) ~ a . ¢(y; a ., c.}m. ~(m./s.)
. 1 l1
1
1
31 1 1
1=
4
+)'2 k
~ a . ¢(y; a .) c. }m . •
o i=l l 1
1
1
1127 and an iterative
Using the tables
such that gl(y}
31
= O.
technique y was found
From this we have the Bayes decision pro-
cedure:
if (x /j2CJ) > (x /j2CJ) - t
l
2
make decision d
if (x /j2CJ) < (x /j2CJ) - t,
ll
1
2
make decision d
and
10
where t is the value of y such that gl(t}
y > t.
(2)
b
l
= b2 = 0
case.
For this case
•
= l-p)
= 0
JJ.
= Yc(c+e)
)
JJ.
= yc(c-e)
)
otherwise.
=0
and gl(y) > 0 for
(,8
and r(y;
£(9) ~ p(l-p) ¢(y; -j2el' 1)
Q)
, Q - - j2ey ,
-LP2+(1-P):7 ¢(y)
= p(
G- 0
Q = j2ey
l-P) ¢(y; j2eT,l)
=0
,
Thus
o
gl (y) - - /2 ko }' (-Q)f(y;
otherwise.
Q)~(Q)dQ
-00
r
00
+ j2k
l
f(y; Q) g(Q)dQ
o
12ko (/2ey)p(l-p) ¢(y; - j2eY,l)
= -
+ ;2kl (/2ey)p(l-p) ¢(y; J2eY,l) .
Thus gl. (y) ~ 0
-¢(y; - J2ey,l)+k ¢(y; j2eY,l) ~ 0
if
that is, if
where k
if
= kl/ko '
Taking natural logarithms, gl(Y) ~ 0
-2 j2yey ~ ln k ,
that is, if
But
Therefore
and
Therefore
if
exp {-LtY+2ey)2+(y-2eY):7/2} ~ k
-y < (In k)/2 j2ey.
var(~) = 4p(1_p)e 2y2rr2 which is to be y 2rr2
e
2
= L4p(1-p17- l
(In k)/2/2ey - jp(l-p) (In k)//2Y .
gl(y) ~ 0
Y ~ - jp{l-p) (In k)/j2y .
89
= jp(l-p)
Let t
(In k)/)2y
and we have the Bayes decision procedure,
make decision d
and make decision d
lO
ll
if (xl//2rr) > (x /)2er) - t
2
(xl /)2rr) < (x2 /j2rr)
if
- t.
The t values for the Bayes decision procedure with the
non-normal prior distribution are given in Table 5 for the
general (b
l
> 0, b
2
> 0) case and the limiting (b l = b 2 = 0)
Fo~ k = 10, y2 = 1 the t values are given for all the
case.
(a3' a ) values of Table 3 while for other k and y2 values the
4
t values are given for a selected number of these (~, a4 )
values.
In Table 5 the t value for a
3
= 0, a4 = 3
is for the Bayes
decision procedure with normal prior distribution.
Comparing
a values with this value of t it
4
is seen that t does vary a great deal with changes in a and a4 ·
3
the values of t for other
~,
However from the illustrations (Figure 8) showing the shapes of
the prior distributions it is seen that the bigger changes in
t are induced by quite large departures from normality.
values for the limiting b
l
= b2 = 0
The t
case are typical examples
of this.
In view of this it would seem reasonable to use the Bayes
•
decision procedure with normal prior distribution if large departures from normality are not suspected in the prior distribution, while if the prior distribution is far removed from normality
the Bayes decision procedure with the more general prior distribution should be used.
90
~le
5
Bayes decision procedure t values
non-normal prior distribution, n = 2, (J02 known
k = 10
2
(a3' '\)
'Y
= ·5
(b 1 > 0, b2 > 0)
(0, 2)
(0, 3)
y
1.562
1.681
(0,18)
1.498
e
1.275
1.414
1.253
(1, 6)
1·355
(1,11)
1.421
(1,18)
1.470
(2
1.346
6)
(2,11)
t
k ::; 1000
2
'Y = 1
1.041
2.433
3·445
1.123
2.584
3·459
1.133
2.402
3.444
1.179
2.597
3.429
1.464
(1, 3)
•
k = 100
2
'Y = 1
1.344
(0,11)
j
k ::; 10
2
'Y = 3
1.266
(0, 6)
e
k = 10
2
'Y = 1
1.689
1.466
(2,18)
1.495
(3,11)
1.542
(3,18)
(b 1 = b2
1.593
=
0)
(1, 2)
1.030
.728
.420
1.456
2.184
(2, 5)
.814
.576
·332
1.152
1.728
(3,10)
.634
.452
.261
.904
1.356
91
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