徐曉嶺, 段貴鋒, 王蓉華, 顧蓓青

(1.上海對(duì)外經(jīng)貿(mào)大學(xué) 統(tǒng)計(jì)與信息學(xué)院,上海 201620; 2.上海師范大學(xué) 數(shù)理學(xué)院,上海 200234)

四單元混聯(lián)系統(tǒng)屏蔽數(shù)據(jù)的統(tǒng)計(jì)分析

徐曉嶺1, 段貴鋒2, 王蓉華2, 顧蓓青1

(1.上海對(duì)外經(jīng)貿(mào)大學(xué) 統(tǒng)計(jì)與信息學(xué)院,上海 201620; 2.上海師范大學(xué) 數(shù)理學(xué)院,上海 200234)

推導(dǎo)了四單元混聯(lián)系統(tǒng)屏蔽數(shù)據(jù)場合下的似然函數(shù),并且給出了常數(shù)失效率單元和線性失效率單元所組成的四單元混聯(lián)系統(tǒng)屏蔽數(shù)據(jù)的參數(shù)的極大似然估計(jì),以及采用似然比構(gòu)造區(qū)間估計(jì)的方法得到參數(shù)的近似區(qū)間估計(jì).

屏蔽數(shù)據(jù); 四單元混聯(lián)系統(tǒng); 極大似然估計(jì); 近似區(qū)間估計(jì)

0 引 言

在可靠性分析中,人們往往通過分析系統(tǒng)的壽命數(shù)據(jù)來估計(jì)該系統(tǒng)中各組成單元壽命分布中的未知參數(shù).系統(tǒng)壽命試驗(yàn)數(shù)據(jù)包括兩個(gè)方面.一是失效時(shí)間;二是失效原因.理想狀態(tài)下,系統(tǒng)的壽命數(shù)據(jù)應(yīng)該包括系統(tǒng)失效的具體時(shí)間以及由哪個(gè)單元失效導(dǎo)致整個(gè)系統(tǒng)失效的信息.但大多數(shù)時(shí)候,導(dǎo)致系統(tǒng)失效的那個(gè)單元并不能夠被準(zhǔn)確識(shí)別出來,人們僅能夠把導(dǎo)致系統(tǒng)失效的原因歸結(jié)為某些單元所組成的一個(gè)集合,系統(tǒng)真正失效的原因被屏蔽掉了.在現(xiàn)實(shí)生活中,由于故障診斷和故障檢測所需的費(fèi)用昂貴,特別是在現(xiàn)代系統(tǒng)中越來越多地采用模塊化設(shè)計(jì),引起系統(tǒng)失效的確切單元通常都是未知的.在對(duì)計(jì)算機(jī)或集成電路等進(jìn)行系統(tǒng)可靠性研究時(shí),也會(huì)遇到相類似的屏蔽問題.導(dǎo)致屏蔽發(fā)生的原因很多,如:經(jīng)費(fèi)的不足、時(shí)間的限制、記錄的錯(cuò)誤、診斷工具的缺乏,及某些單元失效所帶來的破壞性后果等.這使得屏蔽數(shù)據(jù)的統(tǒng)計(jì)分析成為近年來研究的熱點(diǎn)問題,許多學(xué)者做了很好的工作,并取得了一系列研究成果,具體見文獻(xiàn)[1-21].

圖1 四單元混聯(lián)系統(tǒng)

值得指出的是,隨著系統(tǒng)的功能越來越完善,其構(gòu)成也越來越復(fù)雜,例如航空電源系統(tǒng)或雷達(dá)系統(tǒng)等,不再是單純的串聯(lián)或并聯(lián)系統(tǒng),而更多的是多單元的混聯(lián)復(fù)雜系統(tǒng),且常常伴有屏蔽現(xiàn)象發(fā)生.關(guān)于由4個(gè)單元組成的系統(tǒng),除了4個(gè)單元全部串聯(lián)和4個(gè)單元全部并聯(lián)外,系統(tǒng)還有6種不同的構(gòu)成方式,將其統(tǒng)稱為四單元混聯(lián)系統(tǒng).圖1即為4個(gè)單元組成的混聯(lián)系統(tǒng)的一種.

本文作者詳細(xì)推導(dǎo)了如圖1所示的四單元混聯(lián)系統(tǒng)屏蔽數(shù)據(jù)場合下的似然函數(shù),并且給出了常數(shù)失效率單元和線性失效率單元所組成的四單元混聯(lián)系統(tǒng)屏蔽數(shù)據(jù)的參數(shù)的極大似然估計(jì),并采用似然比構(gòu)造區(qū)間估計(jì)的方法得到參數(shù)的近似區(qū)間估計(jì).

1 四單元混聯(lián)系統(tǒng)屏蔽數(shù)據(jù)的似然函數(shù)

在建立模型之前,先給出一些基本假設(shè):

假設(shè)1 系統(tǒng)由4個(gè)獨(dú)立單元混聯(lián)而成;

假設(shè)2 屏蔽的發(fā)生與失效原因和時(shí)間無關(guān)(即獨(dú)立);

考慮將n個(gè)四單元混聯(lián)系統(tǒng)進(jìn)行壽命試驗(yàn).記Tij表示第i個(gè)系統(tǒng)中第j個(gè)單元的壽命,其觀察值為tij,i=1,2,…,n,j=1,2,3,4.則第i個(gè)系統(tǒng)的壽命Ti為:

Ti=min[max(Ti1,Ti2),max(Ti3,Ti4)].

其觀察值為ti,i=1,2,…,n.令Si為引起第i個(gè)系統(tǒng)失效的單元集合,其觀察值為si,i=1,2,…,n.若si中只由一個(gè)單元組成,則表明引起第i個(gè)系統(tǒng)失效的原因是確切的;若si中的單元數(shù)大于一個(gè),則表明引起第i個(gè)系統(tǒng)失效的原因是未知的,即引起第i個(gè)系統(tǒng)失效的單元壽命數(shù)據(jù)被屏蔽了.

現(xiàn)在考慮第i個(gè)系統(tǒng):

因此,

其中,P(ti

于是可以分別給出各單元失效是導(dǎo)致第i個(gè)系統(tǒng)失效的確切原因的概率:

Pi1=P(ti

Pi2=P(ti

Pi3=P(ti

Pi4=P(ti

記

fi1=f1(ti)F2(ti)[1-F3(ti)F4(ti)],fi2=F1(ti)f2(ti)[1-F3(ti)F4(ti)],

fi3=[1-F1(ti)F2(ti)]f3(ti)F4(ti),fi4=[1-F1(ti)F2(ti)]F3(ti)f4(ti),

根據(jù)假設(shè)2,屏蔽的發(fā)生與失效原因和時(shí)間無關(guān),即對(duì)于j,j′∈si有:

現(xiàn)在考慮n個(gè)四單元混聯(lián)系統(tǒng)進(jìn)行定時(shí)截尾壽命試驗(yàn).定時(shí)截尾時(shí)間τ,這時(shí)共有r個(gè)系統(tǒng)失效,其次序失效時(shí)間分別為t1,t2,…,tr.此時(shí)似然函數(shù)可表示為:

其中,C>0為正常數(shù),T為四單元混聯(lián)系統(tǒng)的壽命.

易見,如果考慮n個(gè)四單元混聯(lián)系統(tǒng)進(jìn)行定數(shù)截尾壽命試驗(yàn).定數(shù)截尾數(shù)為r,其次序失效時(shí)間分別為t1,t2,…,tr.此時(shí)似然函數(shù)可表示為:

其中,τ=max(t1,t2,…,tr).

2 單元失效率同為常數(shù)的四單元混聯(lián)系統(tǒng)屏蔽數(shù)據(jù)的統(tǒng)計(jì)分析

設(shè)單元1的壽命為X,單元2的壽命為Y,單元3的壽命為Z,單元4的壽命為K,它們的失效率同為常數(shù)α,X,Y,Z,K相互獨(dú)立,系統(tǒng)的壽命記為T,則T=min[max(X,Y),max(Z,K)].

對(duì)于t≥0,P(T>t)=P(max(X,Y)>t)P(max(Z,K)>t)=e-2αt(2-e-αt)2.

2.1 全樣本場合下的統(tǒng)計(jì)分析

在全樣本場合下的對(duì)數(shù)似然函數(shù)為:

給定不同的樣本容量n,及ni,i=1,2,…,14和參數(shù)真值的情況下,通過1 000次的Monte-Carlo模擬,得到參數(shù)α的極大似然估計(jì)的均值和均方誤差,在給定置信水平0.95下得到參數(shù)α和近似區(qū)間估計(jì)的平均上限、平均下限以及1 000次模擬真值落在區(qū)間估計(jì)外面的個(gè)數(shù),模擬結(jié)果表明極大似然估計(jì)以及近似區(qū)間估計(jì)的精度還是令人滿意的.

例1 取樣本容量n=38,n1=6,n2=4,n3=5,n4=4,n5=1,n6=2,n7=1,n8=1,n9=2,n10=3,n11=2,n12=1,n13=2,n14=2,4個(gè)單元的失效率同取為α=1,通過Monte-Carlo模擬產(chǎn)生38個(gè)失效數(shù)據(jù)如下:

si={1}:2.0327,0.061004,0.35403,0.57617,1.1416,2.1044;si={2}:0.40628,0.19919,1.3348,0.5073;si={3}:0.78044,0.35211,0.6749,1.1586,0.44911;si={4}:0.71715,1.7075,0.86229,0.043607;si={1,2}:0.41627;si={1,3}:0.08604,0.2081;si={1,4}:0.57083;si={2,3}:1.8555;si={2,4}:0.33451,1.9864;si={3,4}:0.42826,0.65081,0.12387;si={1,2,3}:0.51431,1.242;si={1,2,4}:1.0934;si={1,3,4}:1.3014,1.331;si={2,3,4}:1.1799,1.0782;si={1,2,3,4}:1.5512,0.69633.

2.2 截尾樣本場合下的統(tǒng)計(jì)分析

2(n-r)ατ+2(n-r)ln(2-e-ατ).

類似地,可用似然比的方法給出參數(shù)α的近似區(qū)間估計(jì).

給定不同的樣本容量n,和定數(shù)截尾數(shù)r,ri,i=1,2,…,14和參數(shù)真值的情況下,通過1 000次的Monte-Carlo模擬,得到參數(shù)α的極大似然估計(jì)的均值和均方誤差,在給定置信水平0.95下得到參數(shù)α和近似區(qū)間估計(jì)的平均上限、平均下限以及1 000次模擬真值落在區(qū)間估計(jì)外面的個(gè)數(shù),模擬結(jié)果表明極大似然估計(jì)以及近似區(qū)間估計(jì)的精度滿足要求.

例2 取樣本容量n=38,r=36,r1=5,r2=6,r3=6,r4=5,r5=2,r6=1,r7=1,r8=2,r9=1,r10=2,r11=1,r12=1,r13=1,r14=1,4個(gè)單元的失效率同取為α=1,通過Monte-Carlo模擬產(chǎn)生36個(gè)失效數(shù)據(jù)如下:

si={1}:1.4193,0.2747,1.2822,0.8280,0.0654;

si={2}:1.0330,1.3849,1.5964,1.5312,0.5222,1.6606;

si={3}:1.1026,0.4690,0.5532,1.0306,1.0851,1.2903;

si={4}:0.4033,1.7213,0.2125,0.7960,0.9508;si={1,2}:0.7352,0.5709;

si={1,3}:0.6101;si={1,4}:0.7919;si={2,3}:0.3400,0.5037;

si={2,4}:0.6608;si={3,4}:0.8442,0.9803;si={1,2,3}:0.3726;

si={1,2,4}:1.0369;si={1,3,4}:0.7288;si={2,3,4}:1.2013;si={1,2,3,4}:0.8148.

3 單元失效率同為過原點(diǎn)的線性函數(shù)的四單元混聯(lián)系統(tǒng)屏蔽數(shù)據(jù)的統(tǒng)計(jì)分析

設(shè)單元1的壽命為X,單元2的壽命為Y,單元3的壽命為Z,單元4的壽命為K,它們的失效率都為βt,X,Y,Z,K相互獨(dú)立,系統(tǒng)的壽命記為T,則T=min[max(X,Y),max(Z,K)].

3.1 全樣本場合下的統(tǒng)計(jì)分析

在全樣本數(shù)據(jù)下對(duì)數(shù)似然函數(shù)為:

類似地,可用似然比的方法給出參數(shù)α的近似區(qū)間估計(jì).給定不同的樣本容量n,及ni,i=1,2,…,14和參數(shù)真值的情況下,通過1000次的Monte-Carlo模擬,得到參數(shù)α的極大似然估計(jì)的均值和均方誤差,在給定置信水平0.95下得到參數(shù)α和近似區(qū)間估計(jì)的平均上限、平均下限以及1000次模擬真值落在區(qū)間估計(jì)外面的個(gè)數(shù),模擬結(jié)果表明極大似然估計(jì)以及近似區(qū)間估計(jì)的精度滿足要求.

例3 取樣本容量n=38,n1=6,n2=4,n3=5,n4=4,n5=1,n6=2,n7=1,n8=1,n9=2,n10=3,n11=2,n12=1,n13=2,n14=2,4個(gè)單元的失效率同取為βt=t,通過Monte-Carlo模擬產(chǎn)生38個(gè)失效數(shù)據(jù)如下:

si={1}:2.0327,0.061004,0.35403,0.57617,1.1416,2.1044;si={2}:0.40628,0.19919,1.3348,0.5073;si={3}:0.78044,0.35211,0.6749,1.1586,0.44911;si={4}:0.71715,1.7075,0.86229,0.043607;si={1,2}:0.41627;si={1,3}:0.08604,0.2081;si={1,4}:0.57083;si={2,3}:1.8555;si={2,4}:0.33451,1.9864;si={3,4}:0.42826,0.65081,0.12387;si={1,2,3}:0.51431,1.242;si={1,2,4}:1.0934;si={1,3,4}:1.3014,1.331;si={2,3,4}:1.1799,1.0782;si={1,2,3,4}:1.5512,0.69633.

3.2 截尾樣本場合下的統(tǒng)計(jì)分析

對(duì)數(shù)似然函數(shù)為:

類似地,可用似然比的方法給出參數(shù)α的近似區(qū)間估計(jì).給定不同的樣本容量n,和定數(shù)截尾數(shù)r,ri,i=1,2,…,14和參數(shù)真值的情況下,通過1 000次的Monte-Carlo模擬,得到參數(shù)α的極大似然估計(jì)的均值和均方誤差,在給定置信水平0.95下得到參數(shù)α和近似區(qū)間估計(jì)的平均上限、平均下限以及1000次模擬真值落在區(qū)間估計(jì)外面的個(gè)數(shù),模擬結(jié)果表明極大似然估計(jì)以及近似區(qū)間估計(jì)的精度滿足要求.

例4 取樣本容量n=38,r=37,r1=2,r2=3,r3=2,r4=3,r5=2,r6=3,r7=3,r8=2,r9=3,r10=3,r11=3,r12=2,r13=3,r14=2,4個(gè)單元的失效率同取為βt=2t,通過Monte-Carlo模擬產(chǎn)生37個(gè)失效數(shù)據(jù)如下:

si={1}:0.9450,0.6555;si={2}:0.6093,0.6055,0.4182;si={3}:0.3874,0.3345;si={4}:0.2879,1.8960,0.7146;si={1,2}:1.4526,0.6033;si={1,3}:0.5715,0.9877,0.0242;si={1,4}:1.0173,0.5948,1.6265;si={2,3}:0.9140,0.6306;si={2,4}:1.7371,1.0492,0.2713;si={3,4}:0.7639,0.1641,1.0685;si={1,2,3}:1.4754,0.4533,1.1277;si={1,2,4}:1.0529,1.9911;si={1,3,4}:1.0176,0.4860,1.4511;si={2,3,4}:1.3909,0.5437;si={1,2,3,4}:1.0582.

[1] Usher J S,Hodgson T J.Maximum likelihood analysis of component reliability using masked system life-test data [J].IEEE Transactions on Reliability,1988,37(5):550-555.

[2] Doganaksoy N.Interval estimation from censored & masked system-failure data [J].IEEE Transactions on Reliability,1991,40(3):280-286.

[3] Lin Dennis K J,Usher J S,Guess F M.Exact maximum likelihood estimation using masked system data [J].IEEE Transactions on Reliability,1993,42(4):631-635.

[4] Reiser B,Guttman I,Lin Dennis K J,et al.Bayesian inference for masked system lifetime data [J].Appl Statist,1995,44(1):79-90.

[5] Usher J S.Weibull component reliability—prediction in the presence of masked data [J].IEEE Transactions on Reliability,1996,45(2):229-232.

[6] Lin Dennis K J,Usher J S,Guess F M.Bayes estimation of component-reliability from masked system-life data [J].IEEE Transactions on Reliability,1996,45(2):233-237.

[7] Sarhan A M.Reliability estimations of components from masked system life data [J].Reliability Engineering and System Safety,2001,74:107-113.

[8] Sarhan A M.The bayes procedure in exponential reliability family models using conjugate convex tent prior family [J].Reliability Engineering and System Safety,2001,71:97-102.

[9] Sarhan A M.Estimation of system components reliabilities using masked data [J].Applied Mathematics and Computation,2003,136:79-92.

[10] Sarhan A M,Ahmed H.El-Bassiouny.Estimation of components reliability in a parallel system using masked system life data [J].Applied Mathematics and Computation,2003,138:61-75.

[11] Sarhan A M,Awad I El-Gohary.Estimations of parameters in Pareto reliability model in the presence of masked data [J].Reliability Engineering and System Safety,2003,82:75-83.

[12] Sarhan A M.Parameter Estimations in linear failure rate model using masked data [J].Applied Mathematics and Computation,2004,151:233-249.

[13] Sarhan A M.Parameter Estimations in a general hazard rate model using masked data [J].Applied Mathematics and Computation,2004,153:513-536.

[14] Sarhan A M.Bayes estimations for reliability measures in geometric distribution model using masked system life test data [J].Computational Statistics and Data Analysis,2008,52(4):1821-1836.

[15] Awad El-Gohary.Bayesian estimation of the parameters in two non-independent component series system with dependent time failure rate [J].Applied Mathematics and Computation,2004,154:41-51.

[16] Hutto D E,Mazzuchi T,Sarkani S.Analysis of reliability using masked system life data [J].International Journal of Quality & Reliability Management,2009,26(7):723-739.

[17] Tsai H F,Wan L W.Accelerated life tests for weibull series systems with masked data [J].IEEE Transactions on Reliability,2011,60(3):557-569.

[18] Hou H L,Jiang Y W,Shi Y M.Parameter estimations in BurrXII model using masked data [J].Chin Quart J of Math,2011,26(2):251-255.

[19] Xu A C,Tang Y C.Bayesian analysis of pareto reliability with dependent masked data [J].IEEE Transactions on Reliability,2009,58(4):583-588.

[20] Wang R H,Xu X L,Gu B Q.The statistical analysis of parallel system for Type-I censored test using masked data [C].Recent Advance in Statistics Application and Related Areas-2nd Conference of the International Institute of Applied Statistics Studies,Qingdao,China,July 24-29,2009,789-795.

[21] Xu A C,Tang Y C.An overview on statistical analysis for masked system lifetime data [J].Chinese Journal of Applied Probability and Statistics,28(4):380-388.

(責(zé)任編輯：馮珍珍)

Statistical analysis of four-unit hybrid system for masked data

Xu Xiaoling1, Duan Guifeng2, Wang Ronghua2, Gu Beiqing1

(1.School of Statistics and Information,Shanghai University of International Business and Economics,Shanghai 201620,China; 2.College of Mathematics and Science,Shanghai Normal University,Shanghai 200234,China)

The likelihood function of four-unit hybrid system is deduced based on masked data.The maximum likelihood estimates of parameters are proposed for hybrid system composed of four units with constant failure rate and linear failure rate based on masked data.Besides,the approximate interval estimates of parameters are obtained by using likelihood ratio to construct interval estimate.

masked data; four-unit hybrid system; maximum likelihood estimate; approximate interval estimate

2015-07-01

上海市教育委員會(huì)科研創(chuàng)新重點(diǎn)項(xiàng)目(14ZZ155)

徐曉嶺(1965-),男,教授,主要從事應(yīng)用統(tǒng)計(jì)方面的研究.E-mail:xlxu@suibe.edu.cn

O 213

A

1000-5137(2017)02-0178-08