TRU Mathematics 16−2 〔1980〕
THE UNBIASED ESTIMATION OF’IHE PROP（DRTION ON A GIVEN
DOMAIN OF THE MULTIVARIATE NORMAL DISTRIBUTION
BASED ON THE SUFFICIENT STATISTICS
Kenj i MASUDA 〔Received Nbv㎝ber 14，1980）1．INTRODUCTION
LIEB㎜and RESNIKOFF［5］ presented a collection of varia1）1e samPIing Plans based dn the estimator♪ of the per cent defective of a submitted lot． The variable sampling plans pertain to a single quality characteristic， and it is assumed that mea．surements of this quality characteristic are independent ，identically distributed nonnal random variables． Associated with each inspection characteristic are the design specifica− tions・ If UPper lilnitびand lower li皿it L are specified， the item is consid− ered defective if its measurement either exceeds σor is smaller than乙． If the per cent defective of a submitted lot is sufficiently sma11， the lot is accepted， otherwise it is rej ected． S工nce the㎞owledge about the per cent defective is rare， a logical procedure is to estimate the per cent defective from a sa㎎）1e， and accept or reject the lot on the basis of this estimator P． ・ The purPose of this paper is to extend the point estimation theory fbr the mivariate nomal case originally developed by KOI；lvK）GOROV［3］to that fbr the㎜1tiva・i・t・n・㎝・1 case・［El・e・bserv・d v・lues xヱ’・2’…’・。 are ass㎝・dto be independently drawn fr㎝amultivariate nonnal population． The popula−
tion paranleter p to be estimated here is considered to be a probability that an observed value is contained in a specified rmltivariate domain A． As…m exa叩1e， the mifOr血1y mini皿皿variance mbiased estimation in bi− variate nol孤al case is discussed． Nthough this paper deals only with the pomt estlmatlon，1t ls also applicable to the theoエy of testing statistical h）rp《）thesis in sa呵）1ing inspection（see：1．IEBERMAN and RESNII（OFF［5］）． 2．UNIFORMLY MINIMUM VARIANCE UNBIASED〔U．M．V．U．）ESTIMA7て）R 2−1U．M．V．U． Estimator Let xヱ’x2’…’x刀be a random sa叩1e from a k−variate population with density functionψfxノ． The probability p that observed value x is contained i皿 domain A is given by155
156
K．MASUDA
P 一 J。 th （x？． dX’ ・ 『・
Whereρis also cgnsidered as the popuユation paτameter in the fbllowing dis← cussi㎝． Define p as the usual attTibute estimator of the population paraln一 eter P， P（Xヱ’X2’…’Xn，＝ヱ戸f XヱEA’ 〔2．1．1） prx1’X2’・1・・’X刀，＝0’エf Xヱ孔 It is evident that ♪ is unbiased． From I正《and SCHEFFE，s results［4］， itfollows that倉rE‘p／T）is the unique unifomiy血t㎜ variance unbiased esti−
matbT ofρ， 諭ere T is assumed to be the co加plete sufficient statistics fbr the k−variate distribution．2−2Multivariate Nor rnal Distributi㎝ ‘ 、
We shall dhoose the density fimcti㎝〔2．2．1）as basic to；our study of thek−variate nomal distributi㎝，
〔2・2・・〕 …）−k／21・「1／2・xp｛一〉（x一岡ノ’Σ一ヱ（x−Mノ｝， 『． 砲ere Σ is the positive definite matrix． It was also observred that E‘x，＝M andΣ is the matrix of the variances and covariances of the co㎎xcmlents of x． It may be verified that fx，S） is a complete sufficient statistics fbr ・・，・・紬・rrl．、・、紐・・一Σ1．、・・、一叉，・・、一又・・，（see・G・R・［・D．…』・ are mutua11γinφependent〔see：RAO［9］）． ． 2−3U．M．V．U． Estimator −MUltivariate Normal Distribution − If the sa㎎）1e is drawn from a nor皿al population with un㎞own mean vector 珂and un㎞o岨variance−covariance皿atrixΣ， then tlie pair of sample value x and S is complete sufficient statistics． III this case， 〔2．3．1） 食（x．Sノ．＝E（多／x，S／ is the V．M．V．U． estimator fbr the population・parqinetpr p． From 〔2．3．1〕，wehave
（2．3．2） P‘x，S，＝pb｛P・一ヱ／x，S｝ 一臓ii；s戊…’ 油・・efr・ユ｛・’S／i・th・j・int d・n・ity・f・、・・and S…d h（・’S， i・th・j・桓t density Of x and S． Since x and S are mutually independent， and are sul）j ect to a I（−variate −・・d・・t・・bu・i・n V疇・』・・〔tra・…ha・・d・・t・・b…㎝・k・（n一ユ’・1・ respectively（5ee： fbr ex． RAO［9］）， h‘x，S／ is given by、 （2．3．3〕 、 THE UNBIASED EST工MtNTION OF THE PROPORTION hcx’S”＝ C、。」V・Silii／2！z1・／・ex・｛一；’X−『一ヱ剛x ・KlΣ1’ Cn−1）／21S1 （n一ヱーk”1）／2。。p｛一・。。ce（X’ヱS〃・｝， where・K・t…−W…lk−・〃・・：ヱr・・一≒…〃一ヱ． To find．fイxヱ’x／S／ consider the j oint density’of the mutually inde− P・nd・nt sa・XP1・・t・ti・tics xヱ・叉★・nd S★・ 油ere叉・r☆Σ1．、・、．・n・S・一Σ1．、・・ゴ叉・…て又・熾…9…n・by
縮頑一〉‘xヱーM”， z一ヱ（X」”M」｝±㎏｛一ご醐1×
・r。…ヱΣ，一ユr叉・−M」｝・・1・1“’C刀一2，／21S・1 Cn−2−k一ヱ，／2・xp｛一…q・‘・−1S．t ）Z・｝， 曲ere−・〆η一2’・k／2・k・「k・’）／4F1ヱr・n−≒ゴ≠ユ〃−1・・・in・・r・…・・ヱ・S・一⊇・・ゴ又，1・ゴ迦iliiix；iS；；一・・E・，k・
the j oi皿t density f（xヱ，又，S， becomes then 〔・・…）d⊇ゴ・・亡ユ・・ヱーM）・，、。IIラ；llii…
『×expt一字・叉一・，・Σ一ヱ・又一…。…、・ヌーMl・・一ヱIR−Xヱ・・。…、・又一X、，・・ ・Σ一ヱ（R−M）・＋15（叉一Xヱ，・・一ヱ・叉一Xヱ，｝・・ （n−1， …1・1’・（n’2）／21S−U1‘n−2−k−”／2・・pt−’・…e｛・一ヱ・S−U，｝／…。三、，k， Where U−。i、‘Xヱー叉，r・ヱー叉，’・ The fbllowing equation is evident： lS−。三、（・ヱー又ノ‘・ゴ又ノ・1−IS｜｛・一☆・ヱー叉，・S一ヱ（・ヱー叉ル Since S and S★are nomnegative’definite， respectively， the above equation麺1’es that
Dヱー ￨、・・、一叉，・S一ヱ・・、一又…． Dividing the expression 〔2．3．4） by 〔2．3．3〕 resUltS in the condi．tiona1 −density of x1・ given x and S；157
L158
K．mSUDA
〔2．3．5） f（・ヱ’x’S， hcx，S， ・k／2・‘n G1， 1，、・。三、，k／2・xp・一 ll・、一・…−1・・、一・， n−2−k＋ヱ ）‘2π， rr 2 一垣一Xヱ）s・−1励・IR−M…−11R−X、1・。…、IR−Xユ，’・一ヱcR−・ヱ・｝ …ace・・一ヱU，…÷1衰一・，・Σ一1・又一・〃嶋・We can evaluate the power y of exponential in〔2．3．5〕⊇e允11㎝血g㎜er，
・一一 A間｛・・、ヱー・i，・・j、−U、鴎一X、ユ，・吾戊川三、−Ui，・・デ・戊、， ・。…、ll、一・、ヱ・庁・ゴ、・一。三、・・i、・1，・・j、−iY−・fli−v、，1 ij−・ゴ，｝ ＝0 ． It should be noted here that the fbllowing relations are used， nalnely， 叉一・三、i、＿i。7’，・、一奈；．、・、、’・一・，・，…，k，・−1−・・’jl’・’・一・’・’…・k・ M−‘‘・ヱ・2…P。）’・U−（・、戊，’・げ。≒f三、一・i、）（ ij−・」、ノ’i・」一・’・’…’k and xヱ＝rxヱヱx2ヱ…Xkl」’・ S血ce lS−Ul−IS’1｛ヱー。三、・・、一叉，・S一ヱで・、一又，｝，i・f・…ws・h・・ 〔・…6〕 ﾌノー，，睾・；lii・7・・2・・i・1・／・1，1・／・・ 刀一2−k一ヱ ・｛ユー。三、‘・ヱー又）・S→ユr・ヱー衰，｝2 We・・tice th・t S−1−D（・／屋）R一ヱ・r・／儒，，油・r・Ri・the s卿・e c・rre−i：轡竃璽議i：三竃鐘：蓋欝ぎ
輌g・na・・ma・・iX・D・・、，，噸・・、’i−・，・，…，k ar・th・ ・ig・nv・・ue・・f R一ヱ・ ・f・th・t・an・f・一・i…一・・可P・・／認・・、一叉・i・．ma・・，血・n・th・right me血ber of〔2．3．6）bec㎝es ユユ ’
n−2−k一ユ ．，，；睾・，緒・・1，1・／・1・−i・・2：’：・
THE UNBIASED ESTIMATION OF THE PROPORTION
1・1一篇㌫曇痛｛㌔1、縫玲芸・
then the dens ity of z， given イx，S，，beco皿es as fbllows： 11一ユ 刀一2−k−1毒；・，南輌2 −・h・・rest・…i・n・・ノー
We shall fb㎝…江ize the…ibove result in the fbllowing theorem． 田…OREN・・亡Nk （M’Σ） b・亡力・k−va・i・亡・・。rrn・ヱP・puユa亡i°n and xヱ’x2’…’X。 加arand。m sample fr。m the a加▽e n・rmaヱP・Pα1a亡ion・ Then，亡力e unigue uni− forml y皿inimun vaヱ・iance unbiased es亡ゴma亡oヱ」）（x，S， of 亡he probabiユi亡y P tha亡 亡he observed value X is oo刀亡ainedゴηadomain A is give刀as fbヱユows：（・・…）・（R．s）＝IT」A，，；睾己ピ芦三
n ユ ーぬ・・e・、A−｛・−Df万フP・r Atransfonnation T 2 Where z＝aσOSθ ユ’ フ ユthe transformation T
2， COROLLARY． Fo r k≧2 n一ヱ is defined as 戊一ヱ z戸aσosθS∬呈＝1 〔2・3．7〕 ．一戊（・XゴX）／・ユ・A｝・ ユユ T2r・ユ・2…・k，＝「aθ、θ2… s・n・、，戊一・，・’…’k−・・and・。一・己二；・i・・、・ is presented as following corollary・ θ ，， k一ユUnder
一⇒。。毒iと，が已瓢二i・・…i・k−i−ideide・
2ユ …dek．ヱda’、 』e…・己二i （Sin・ノー’−1−・ ・・r k−一・・3．EXAMPLE
We assume k＝2． Then we substitute in 〔2．3．8）．（3・・）P・R…
ﾟ÷認…・
He・eafter，1 GS・3‘ ；1ユーa2パη一5”2 nt・・b・d・・…dby・1・）・・mce the c・rre・a−・i・一・…bec・−R−
m：；］・’・・ h・v・ R一ユー、i｝。T・［．司・・…ig−・・ue・ ・、，・2・fR一ヱare・b・・垣・d・λユ☆・・2−、…ピ159
160
K．MASUDA． The。rth。g。n。・ma・riX・P・whiCh・・an・f・rm・R一ユt・・（・i，’・一・’・i・gi・・n・by ・一m；傷倒・
Fro皿 （2．3．7） fbr k＝2， it is easily seen that［；；］輌・・馬Cl：il］・
The above relation is rewritten as fbllows：（…）・2一癌ユー岳号嘉・
（…〕・2−一震、・繧乏据・
ロ The data below is a random sa匝ple of sizeユo ta］（en from a bivarlate nor一 皿al populati㎝・ ．ユ ．ユ，ユー2
ヱ 2 3 4 5 6 7 8 9 ヱ0 8．639 8．634 9．ユ48 9．057 9．757 9．028 10．336 7．978 9．693 8●703 9．76 9．68 7．88 9．50 8．6ユ 8．55 ヱ0．63 7．38 9．98 9．ヱ5 S・rPP・・e th・t・pecificati・・liiT！it ・、一ユ…f・r・、ヱ・nd・、＝ヱ゜・°f°「x2ヱ・Now，。e・h・11 estim・t・th・per cent d・fectiv・pf・・th・ab・ve・pecificati㎝
li皿its． D・fi・・A−｛↓・、、・、、〃（・ユユ・u7・r・、、・・，，｝・F…〔3・2〕and〔3・3）・the T・A in （2．3．7〕 is given by（・．・〕…、・2〃・癌、一岳綜ユk・21・・一掛岳鰺2綜
・・Q川・i・・Z・ヱル We shall denote （3．4）， for s iirrplicity， by T、A−…、・2〃・σzヱ・・、・・21・・一・・、・・、・・21…i・・；・ヱ・｝’wh・re lil／ff・・、酷鰺矯・・d・、嘉鷲鳥
・・nce・、一一・．・4・892，・、一・・928478・…一・・36・72・w・・ha…bi・・…2，・ヱ／2− ・．・・5998，・b；・・…2，・ユ／2−・．362ユ5ヱ…ケ・・、一・、12／・2…、・・、12・ユ／2・・…42・882・ TAヱTA2 TA3 and r4 ・r・giv…sf°ヱユ゜WS； TA、一・、・、A・…，・，・・2・・；・…σ21｝・TA、一・、・、A・…’・〃・；・1…21＜a2・・i・・…21｝’TH巳UNBIASED ESTIMAITION． OF TH巳PROPORTION 盟，−T、・、A・｛〔・’・〃・：／…ろ・・2・i・ （b、一・ジ／・2・（b、・・、）2J｝and TA、−v・｛‘…，／i・‘・ヱー・、，2／・2・・（b、・・ジ1・a2・・｝，・ee・ig・…，wher・’一・th・ t・anSf・nnation T2 i・giV・n by ・2r・ヱ・ノー（・’θノ…θ・2…ヱ篭…θ・・d・2⊇⊇θ・ 2
