定時(shí)截尾下具有缺失數(shù)據(jù)兩個(gè)Rayleigh分布總體參數(shù)的估計(jì)和檢驗(yàn)

趙志文,M.S.Abdalroof,盛丹姝

(1.吉林大學(xué) 數(shù)學(xué)學(xué)院,長(zhǎng)春 130012;2.吉林師范大學(xué) 數(shù)學(xué)學(xué)院,吉林 四平 136000)

數(shù)據(jù)缺失問題是統(tǒng)計(jì)學(xué)中的常見問題,如在產(chǎn)品的壽命實(shí)驗(yàn)中,由于觀測(cè)方法、實(shí)驗(yàn)設(shè)備或其他原因常會(huì)導(dǎo)致某些實(shí)驗(yàn)觀測(cè)數(shù)據(jù)缺失,因此對(duì)不完全數(shù)據(jù)的處理是統(tǒng)計(jì)學(xué)的一個(gè)重要研究領(lǐng)域[1].此外,在可靠性壽命實(shí)驗(yàn)中,為減少人力、物力及財(cái)力的浪費(fèi),實(shí)驗(yàn)者常會(huì)采用定時(shí)或定數(shù)截尾實(shí)驗(yàn)[2-4].當(dāng)產(chǎn)品的壽命分布為Rayleigh分布時(shí),文獻(xiàn)[5]討論了具有缺失數(shù)據(jù)的兩個(gè)Rayleigh分布總體參數(shù)的估計(jì)問題及兩總體參數(shù)相等的假設(shè)檢驗(yàn)問題.對(duì)于定數(shù)截尾數(shù)據(jù),Harter等[6]給出了未知參數(shù)的極大似然估計(jì);Howlader等[7]進(jìn)一步討論了未知參數(shù)的Bayes估計(jì)及未來(lái)觀測(cè)值的預(yù)測(cè)問題.對(duì)于步進(jìn)刪失樣本,Wu等[8]考慮了未知參數(shù)的Bayes估計(jì)及未來(lái)觀測(cè)值的預(yù)測(cè)區(qū)間問題.本文在此基礎(chǔ)上進(jìn)一步討論定時(shí)截尾下,具有缺失數(shù)據(jù)的兩個(gè)Rayleigh總體參數(shù)的極大似然估計(jì)及兩個(gè)總體參數(shù)相等的假設(shè)檢驗(yàn)問題,給出了參數(shù)的極大似然估計(jì)量,并證明了估計(jì)量的強(qiáng)相合性、漸近正態(tài)性及檢驗(yàn)統(tǒng)計(jì)量和檢驗(yàn)統(tǒng)計(jì)量的極限分布.

1 極大似然估計(jì)及其漸近性質(zhì)

下面考慮參數(shù)λ1的極大似然估計(jì).在得到觀測(cè)值(Zi,δi,αi)(i=1,2,…,n)后,相應(yīng)的似然函數(shù)為

其中:Ai=αiδi(αiδi+1)/2;Bi=αiδi(αiδi-1)/2;i=1,2,…,n.進(jìn)一步,取對(duì)數(shù)有

同理,基于樣本觀測(cè)值(Mj,ηj,βj)(j=1,2,…,n),可得參數(shù)λ2的極大似然估計(jì)

(1)

其中:Cj=βjηj(βjηj+1)/2;Dj=βjηj(βjηj-1)/2;j=1,2,…,n.

證明:由于{αiδi,1≤i≤n}為獨(dú)立同分布的隨機(jī)變量序列,故由強(qiáng)大數(shù)定律知

由Slusky定理可知

由引理1可知

其中

2 兩個(gè)總體參數(shù)相等的檢驗(yàn)及兩個(gè)總體參數(shù)之差的漸近置信區(qū)間

在實(shí)際問題中,人們通常關(guān)心兩組樣本是否來(lái)自同一個(gè)總體,該問題可以歸結(jié)為假設(shè)檢驗(yàn):H0:λ1-λ2=0 ?H1:λ1-λ2≠0.

(2)

特別地,在原假設(shè)H0下,有

(3)

證明：由命題1和命題2可知

由Slutsky定理可知式(2)成立.證畢.

因此,對(duì)于給定的置信水平α,Δλ的置信區(qū)間為

3 隨機(jī)模擬

表1 n=50時(shí)估計(jì)的偏差和覆蓋率Table 1 Estimate bias and coverage probability when n=50

表2 n=100時(shí)估計(jì)的偏差和覆蓋率Table 2 Estimate bias and coverage probability when n=100

表3 n=300時(shí)估計(jì)的偏差和覆蓋率Table 3 Estimate bias and coverage probability when n=300

由表1～表3可見,無(wú)論對(duì)于較小的樣本量還是較大的樣本量,本文的估計(jì)方法都有較小的誤差,并且|Δλ|的置信區(qū)間覆蓋率非常接近置信水平0.90,表明本文方法具有較高的精度.

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