A BAYES MULTIPLE DECISION PROCEDURE FOR SELECTING THE BEST ONE AMONG SEVERAL NORMAL POPULATIONS WITH COMMON KNOWN VARIANCE

 Ａ BAYES

MULTIPLE

DECISION

PROCEDURE

FOR

SELECTING THE BEST ONE AMONG SEVERAL

NORMAL

POPULATIONS

WITH

COMMON

KNOWN

VARIANCE

       By

Yukio

NoMACH and

TaKashi

Matsuda

       (Ｋｏｃｈｉ

ｕｕi。ｅｒｓｔり)

 §1 . Summary. A problem which selects the best one among several assigned normal populations with known variances, has been discussed by many authors such as Bechhofer, Ｒ. Ｅ，〔1〕, Paulson, E.〔2〕, Nomachi, Y.〔3〕and many others.

 The purpose of this paper is to present ａ Bayes procedure for selecting the best one among k normal populations with common known variance; depending on ａ ｐｒioｒｉsample means. The mathematical model appears in quality control and analysis of variance model

and etc.

 In order to progress our comprehensions. let us considｅｒ k processes producing items

continuously. In this case, it is often desired to chose the best lot (the “best" lot is the lot whose percentage of defective items in it is of least among these 1 lots), or to ranking k lot means in the ascending order of magnitude.

 In what follows, foｒ丘xed n and r, let us state the sampling scheme as follows :

 (1) For each f in 1≦f≦k, at the ･firststage of sampling, it is assumed that the unobservable sample mean μi is drawn from ａ ｐｒioｒｉnormal population Ｎぐジ(゛)，72)，

where 72 is known, but y<" is unknown to us. Then the observable sａｍｐｌｅ{刈外麗,2>･ ･ ･ >涙声

is drawn from the normal population ｙ(μび＼ a^), where d' is known to us. Then let us

go to the next stage of sampling, and so on.

 (2) For each i in 1 ≦i≦k, at the n-th stage of sampling, it is assumed that the unobservable sample mean μび)is drawn from the normal population ＮＧﾉ(馬？).Ｔｈｅｎ

the observab】ｅ sample ＼Xnl,Xn 2> ■■

■' Xnひ}iS drawn from the normal population ｙ(μび)， ♂).

 (3) For each f in 1≦f≦ん at the (w+l)th stage of sampling after the sample

{ヱ沢1,1･ｊλ?l,2･‥・・Ｊ騨1,ｒ} wasdrawn from the normal population Ｎ(μi+＼. rO, we want to rank unobservable means G=l,2,...,/b) in the ascending order of magnitude or to choose

the largest mean among them.

 In what follows, let us define the (7z＋1)-tｈ means by

   丿心おＪ沢1,φ^‘G-=l,2,…ぶ.

54

高知大学学術研究報告  第19巻、自然科学  第５号

Ｓ 。

丿゛)り= l,2,...,'z+l).       ＼ ニ‥‥‥‥

  Let us assume here that the unolﾆjservable variablesμ■r(j°1,2,…ｎ＋1)ａｒｅ indepen-      ●  4    =jdently and identically distributed according to the commor! ｎむt祐ａ! distribution Ｎり(ｏ，？). Therefore all prior observations 3'" (;'―1,2,...う'+1) can supply certain informations        r.  r ，

tｏμ沢1 through the distribution Ｎ(ν(ｏ，ｒ2)，Ｇ〒1,2.･…, /c) respectively.

  It is our object in the next section to present ａ Bayes procedure for ranking means .μn+1 ＼t―l> i.,

■■■,た), depending upon the ｐａＳいz observations

ｙﾘI･(j＝1，2，…，77･ ； j=l,2,        f  ● J

・‥べ), and to present certain numerical comparisons between our results and Bechhofer's  〔1〕. These comparisons n!ay recommend us to t仙ｅ･prior observations in the cases when       ● .  1our mathematical model is applicable.      ノ    ．，

 §2. Notations and definitions.  In order to develope our theory､. let us prepare the following preliminaries.       卜‥‥‥‥｀

 Notations l. Let us write ａ vector in k space Q by   ＼ ．

      μll+＼ ＼μ扁、μ認1、…、μ以)．  卜    卜

 For any set of non-｢legati゛ｅ ゛ａ｣ues of ∠1(幻[t],2 ‘1?，･let:us define such three types

of set of μ741 that      ' ゛'・･∧ ， and 、that 訳μ。Ｊ)＝り411μ臨1)十ｊ“)＝μ沢1G＝2｡し。。、ん)}        ､、    で 、 ￡?(j)＝{μ。1jμ路1}十Ｊ“)＝μ沢1G＝､2よ‥。諌)} l^n+l― max Un+l  ･･････････････････

is true, whenever

鶏剽＝ｍａχ八雲1  ………゛………・…・・‥…･･    1≦f≦ゐ       ゛ ト  ト (1) (2) (3) ( 4 ) ， (5)

￡?(ｊ(恍)＝(μ。njμ気1)＝μ訊?lfor j≠i andμ気1)＋ｊﾂﾞｰ=μ沢1}

 Definition

of decision rules.

Ｌｅtj沢l be certain･ estimate ofμ沢l based on

observa-tions drawn

previously、f＝1、2、‥。ｋ      ＼    ゜＼    ／

respectively. Let us define that        ’゛  、

holds true. The event 芦沢1＝μ以?I G≠j) is an ｅｙｅ叫of probabiりty zero and we may ignore

in probability calculations. However> this event may occur' in ･the actual applications. If        I･      ● g

this event occur the tied means may be ranked by means of ･a randomized technique which        ● y     j  ｀

assigns equal probability to each ordering. Let us say s皿h event that the relation (4) holds

true when the relation (5) is true by the correct selection (CS) of means.       ¶.

 Notation II. Let {CSLQ(･)} be the event b卜correct, selection of the best mean in the case when■Ｉｎキｘ（Ｑ(･), where we define that the best meanﾆis the maximum mean among        .  :    ｢ゝ 1･

A Bayes multip】ｅdecision procedure for inc the best: one (6) (7) (8) (9) ㈲ 皿 ） 陥 55

 Lemma １･ Under the assumptions state【1 previous】y for the distridutions of μf and

y'P (;･= !、2、…、7z＋1；ｉ＝１、２、‥．、ゐ)、wehave that for each むｎ１≦ｆ≦ん、the conditional

distribution of μ訟1、given ｙ碧いS ･the normａ＼ ａ ｐｏｓtｅｒioｒi distribution

Ｎ(り'万引2y(O、☆)、…………

where Ｅ＝の/ｒ.

 Proof. We

can directly obtain the result from

the conditional density function by use of

the properties ･of normal

density functions.

(e.g. p. 380 in 〔4〕）

 In Lemma

1, hoｗｅver, we cannot determine the normal ａ posterio元distribution, because

we assumed

that the value of ν（ｏＧ＝1，2，‥．諌）ｗｅｒｅ

unknown

to us.

 In this case we will use the usual estimate

ｙ沢1°Ξ1 y

0U-) =

for v'"G=l、2、‥．、ｋ)、respectively. Then we have the fol】owing lemma :  Lemma 2. For each z in 1 ≦f≦応、let us put that

司□1＝（ぴ沢1十心訪1）/（

where we put that

         °゜aiTand夕雲1°Rjy

then戸沢1 has the norn!al distribution

くご,ヅレ(1＋万謡Ｕ))

 Proof. We can directly.〔pbtain the result from the property of normal density functions  (e.g. p. 382 in 〔4〕)

 Lemma ３. Let χ(1)，χ(2)，...，χ(幻ａｒｅindependent and normally distributed.ＮＩＤ(μ(幻 a') (i = 1, 2, . .., ^), random variables. Then for any real u we have

p＼ max X<"≦djl゛(('‘￣岬)ん)

where for the standard normal

density function 心）ｗｅ have put

 Proof. For any real ｕ、since χ(゛j has the distribution Ｎ!Ｄ(μ(゛J、元)(i＝1、2、…、り

we have

56 高知大学学術研究報告  第19巻  自然科学  第５号

= 77 p＼x^勾≦４

  ｌ       ＿ ゜だ 1 のcu-μ(ｏ)/(7).   ゜゛゛･･･i.･･..･..･...･...

Lemma ４. Under the same conditions as in Lemma ３， we have

where ﾊﾞGI°(∼)ト仁 ｆ  ( ぷぷ ４ ﾄﾔ゛-･'ﾚ･     -μ(2･1)/，/Fc，-μ(*,*-!)/ソΓι7 ' ニ が゛り2、ｙ3､‥べyｌ)、jg＝(μ(2･1'、μ(3･2j、....､μ(､゛･゛-1)、が＝(μ−μ)'/、/Ta jyj°χ(3J−χ(J￣IJ、zj(3･J-D ―μり)一μり-1)、り＝2、3、‥.ﾌﾟ､た)

and where Ｐ＝｛ρり}denotes the k-l

by k-[

correlation matriχ with

∼=｛ににぢ (i.

7 = 2,3,･.･,幻

Proof. By virtue of the transformations of variables

yjJ°χり)−χ(J-1?

sり=

￨

にぎ1 21 a3) ㈲ 皿 (16) (摺 旧 ( 2 0 ) 屈 the joint probability density function of y'" (/=2、3、…、k) is given by

y(jy(2J' ｙ3'¨‥'ｙｉ')゜(21万(に:;;2e｀p{￣￣}(ﾀ￣７μｙΣ-Ku-t″)}'…… ｕ

where V and fi are given ｂｙ凶、ａｎｄΣ＝｛ｊり｝iｓthe variance､covariance matrix ofｙ、（j＝ 2、3、…、k) with

i=j

li−jj＝1   （f，に2,3,…ｊ）  ………

O for ＼i-j＼≧2

Therefore we obtain       Ｐ{ＣＳＩ斑j}}＝j）{χ(1)＝χ(2)＝…＝χ(゛'φ(μI)}       ＝j){ｙ(j)≧O，j°2,3,…，創衣ｊ)}  づ ‥・  j’   (2 がし ９ｐトヤ９）一蜘゛ -μ(2･1)/へ/To -芦りIc-1)/､/2a

心＝け2と閲（i,

/■

= 2,3

k)

where we put *'=Cf/-/)'/、/了とy、and P={Pij} is the ん−l by 1 −１ correlation matriχ with

一 一

于

０

A Bayes multiple decision for selectin the best one

昌

(23)

叫

57

 Theorem.

Under

the same

conditions and the same notations as in Lemma

３ and Lemma

４、we have

      P{CS＼Q(μ。。1)}≧？{Ｃ別訳ｊ}}≧ＰｉｃｓＩ訳ｊ(゛)}

( 〉 く ) 戸ＬＬ∼ 于 -ｊ(ね C 〉 く )        ど1゛      (2π)(゛-1μ2 exp )/ｙΓι7＊

where

P was given by (21)and

″＊2°☆｛1十ペニ1｝｝

一十がP-H

c＝ａ/Ｔ

jf、 ‥‥‥

 Proof,

Substitutingがμl given by (8) into jむ田G-=l,2,…,k),

respectively, and c* given

by (23)intoぴused

in Lemma

３ and Lemma

４ the resulting relation(23)directly follows.

 Corollary.

Under

the same

conditions and the same notations as in Theorem,

for any

preassigned valus ofαin

O くα≦I/A,

there exists such unique value of ｊ(幻(α)that

      j）{ＣＳ１２(ｊ(゛J(α))}＝1−α

holds true.

 Proof、 The function Pics￡?(∠1(゛J)ｏｆ non-negative value of ∠1(i･is monotone、 contiouous、

increasing and bounded above by one. In special case ｗhe､ｎ we know that ∠j(゛)＝O、the correct selection of maximum mean is attained by use of certain chance mechanisum which assigns equal probability ＼lk to respective selection of mean. Therefore for any preassigned value of αin O くα≦1μ。

     ＰｉｃｓI∂(j(゛)(α))}＝1−α

holds true.

 § ろConclusion If

we are in the position where ａ ｐｒioｒi

informations are available

as was in our paper、it is recommended to

use them

all insuch ａ manner

as was stated

by us.

 While

the numerical

calculations of the result of the theorem in this paper are preparing

now.

      References

〔１〕Bechhofer, R. E. : A ｓinｓle-ｓａｍがｅ。・ｌｕltｉｐｌｅ ｄｅｃｉｓｉｏｎ夕ｒｏｃｅｉｕｒｃ ｆｏｒ ｒａｎｋｉｎｇ ｍｅａｎｓ ｏｆ ｎｏｒｍａｌ  ＊ｏｌ>ｕlatｉｏｎｓ ｗitｈ ｈｎｍｘｍ ｌ，αΓｉａｎｃｅｓ．Ann. Math. Statist., 25 (1954), 16-39.

〔2〕Paulson, E.:Ａ Ｓｅｑｕｅｎtial pｒｏｃｅｄｕｒｅ ｆｏｒ ｓｅｌｅｃtｉｎｇ tｈｅ ｐｏｐｕlatｉｏｎ ･ａｎｔh. the laｒｇｅｓt ｍｅａｎｓ  fｒｏｍ ｋ ｎｏｒｍａｌ ｐｏｐ ｢ａtｉｏｎｓ， Ann. Math. Statist. 35 (1964), 174-180.

〔3〕Nomachi, Y.:Ａ ｃｌｏｓｅｄ ｓｅｑｕｅｎtial pｒｏｃｅｄｕｒｅ ｓｅｌｅｃtｉｎｇ ｔhe beｓｔ ｐｏｐｕlatｉｏｎ ｉｎ ａ∫ａＴｎｉｌｙ ｏｆ  ｐｏｐｕlaｔｉｏｎｓ ｖoitｈ０ｎｅ ｐａｒａｍｅtｅｒ ｅエｐｏｎｅｎtial diｓtｒibｕtｉｏｎｓ，Bull. Math. Statist., 12, No. 3-4,  21-34.

〔4〕Deely> J. J. and Zimraer, W. J.:Ｓｈｏｒtｅｒ ｃｏｎｆｉｄｅｎｃｅ ｉｎtｅｒｖalｓ ｕｓｉｎｇ ｐｒioｒ obｓｅｒｕａtｉｏｎｓ,  Jour. Amer. Stat. Assoc., 64 (1969), 378-386.