二項(xiàng)分布參數(shù)多層Bayes和E Bayes估計(jì)的性質(zhì)

王建華,毛娟

(1.華中科技大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,湖北武漢 430074;2.武漢理工大學(xué)理學(xué)院數(shù)學(xué)系,湖北武漢 430070)

二項(xiàng)分布參數(shù)多層Bayes和E Bayes估計(jì)的性質(zhì)

王建華1,2,毛娟2

(1.華中科技大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,湖北武漢 430074;2.武漢理工大學(xué)理學(xué)院數(shù)學(xué)系,湖北武漢 430070)

討論無(wú)失效數(shù)據(jù)下二項(xiàng)分布參數(shù)E Bayes估計(jì)和多層Bayes估計(jì)的性質(zhì),證明二項(xiàng)參數(shù)的多層Bayes估計(jì)和E Bayes估計(jì)漸近相等,且E Bayes估計(jì)值小于多層Bayes估計(jì)值.

二項(xiàng)分布；參數(shù)估計(jì)；E Bayes估計(jì)；多層Bayes估計(jì)

1 引言

在可靠性試驗(yàn)中,對(duì)高可靠性產(chǎn)品進(jìn)行定時(shí)截尾試驗(yàn),在規(guī)定的試驗(yàn)時(shí)間內(nèi)往往沒(méi)有樣品失效,獲得的數(shù)據(jù)為無(wú)失效數(shù)據(jù).基于無(wú)失效數(shù)據(jù)的可靠性參數(shù)估計(jì)對(duì)高可靠性產(chǎn)品的可靠性研究具有重要的理論和應(yīng)用價(jià)值[14].韓明博士對(duì)這一問(wèn)題進(jìn)行了系統(tǒng)的研究,所作專著《基于無(wú)失效數(shù)據(jù)的可靠性參數(shù)估計(jì)》是這一研究成果的總結(jié).遺憾的是關(guān)于無(wú)失效數(shù)據(jù)的可靠性參數(shù)多層Bayes和E Bayes估計(jì)性質(zhì)的三個(gè)命題(文[1]中命題2.1,命題3.1,命題6.1)未能給出數(shù)學(xué)證明,只給出數(shù)值算例說(shuō)明.本文給出命題6.1的數(shù)學(xué)證明,其他命題的數(shù)學(xué)證明將另撰文給出．

設(shè)某產(chǎn)品的壽命分布類型是未知的,現(xiàn)從中隨機(jī)抽取n個(gè)樣品進(jìn)行定時(shí)截尾試驗(yàn),若在截尾試驗(yàn)時(shí)間段內(nèi)有X個(gè)樣品失效,又產(chǎn)品的失效與否是相互獨(dú)立的,則X是一個(gè)服從二項(xiàng)分布的隨機(jī)變量,于是有

其中0＜R＜1,R為產(chǎn)品的可靠度.

這樣研究可靠度的非參數(shù)估計(jì)問(wèn)題,就轉(zhuǎn)化為研究二項(xiàng)分布(1)中參數(shù)R的估計(jì)問(wèn)題. 若R的先驗(yàn)分布為Bata分布,其密度函數(shù)為

當(dāng)0＜b＜1,a＞1時(shí),π(R|a,b)為R的單調(diào)增函數(shù),滿足多層先驗(yàn)分布增函數(shù)構(gòu)造法要求.尾部越細(xì)的先驗(yàn)分布會(huì)使Bayes估計(jì)的穩(wěn)健性越差,因此在0＜b＜1時(shí)a不宜過(guò)大,設(shè)a的上界為c(c＞1為常數(shù)).超參數(shù)a和b的取值范圍為區(qū)域D={(a,b)|1＜a＜c,0＜b＜1}. 設(shè)a的先驗(yàn)分布為(1,c)上的均勻分布,b的先驗(yàn)分布為(0,1)上的均勻分布,則在a和b獨(dú)立時(shí),R的多層先驗(yàn)密度函數(shù)為

R的多層Bayes估計(jì)為[1]:

定理1對(duì)二項(xiàng)分布(1),在無(wú)失效數(shù)據(jù)情形下,若R的多層先驗(yàn)密度函數(shù)由(3)給出,則在平方損失下,R的多層Bayes估計(jì)為

證明見(jiàn)文[1].

R的E Bayes估計(jì)為[1]:

定理2對(duì)二項(xiàng)分布(1),在無(wú)失效數(shù)據(jù)情形下,若R的先驗(yàn)密度函數(shù)由(2)給出,則有(i)在平方損失下,R的Bayes估計(jì)為

2 引理

引理1n為正整數(shù),a和b為實(shí)數(shù),當(dāng)0＜b＜1,a＞1時(shí),有

證明文[5]等證明了兩個(gè)Gamma函數(shù)比的不等式

3 定理的證明

[1]韓明.基于無(wú)失效數(shù)據(jù)的可靠性參數(shù)估計(jì)[M].北京:中國(guó)統(tǒng)計(jì)出版社,2005.

[2]韓明.無(wú)失效數(shù)據(jù)情形可靠性參數(shù)的置信限[J].工程數(shù)學(xué)學(xué)報(bào),2004,21(2):245-248.

[3]韓明.無(wú)失效數(shù)據(jù)情形可靠性參數(shù)的估計(jì)和調(diào)整[J].應(yīng)用數(shù)學(xué),2006,19(2):325-330.

[4]韓明.Pascal分布的參數(shù)估計(jì)[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2006,22(4):510-515.

[5]Elezovic N,Giordano C,Pecaric J.The best bounds in Gautschi’s inequality[J].M ath.Inequal.App l.,2000, 3(2):239-252.

The property of hierarchical Bayesian and E Bayesian estim ation of binom iald istribu tion’s param eter

WANG Jian-hua1,2,MAO Juan2

(1.School of Mathem atics and Statistics,Huazhong University of Science and Technology,Wuhan 430074,China;2.Department of Mathematics,College of Science,Wuhan University of Technology,Wuhan 430070,China)

This paper researches the asym ptotic property of hierarchical Bayesian and E Bayesian estim ation of binom ial distribution’s param eter in zero-failure data.This paper gives theMathem atical p roof of that the E Bayesian estimation of binom ial distribution’s parameter is asym ptotic equal to its hierarchical Bayesian estim ation and E Bayesian estim ation of binom ial distribution’s param eter is sm aller than its hierarchical Bayesian estim ation.

binom ial distribution,parameter estimation,E Bayesian estimation,hierarchical Bayesian estimation

O212.8

A

1008-5513(2009)02-0223-08

2007-10-25.

王建華(1965-),副教授.研究方向:應(yīng)用概率統(tǒng)計(jì)與金融數(shù)學(xué).

2000M SC:62F12,62F15