在競爭風險模型中, 該物件中的任何一個風險因子失效皆會導致物件失效。由於這些因子來自於同一個物件, 所以因子間可能是相關的。在本篇文章中, 我們考慮在混合設限計劃下, 風險因子具二元Marshall-Olkin 指數壽命的競爭風險模型。混合設限計劃綜合了型一和型二設限計劃, 是壽命試驗或可靠度分析中常用的方法。而在競爭風險模型中, 時常無法得知引起物件失效之風險因子, 這類資料稱為隱蔽資料。我們使用期望值-最大化演算法去求得模型中參數之最大概似估計, 並以遺失資訊法則估計其標準誤。另外我們也分別使用使用主觀先驗分佈和無訊息先驗分佈, 由馬可夫鏈蒙地卡羅方法得貝氏估計。並比較兩種方法所得模型參數以及物件和風險因子的平均壽命和可靠度之統計推論。模擬結果顯示, 當隱蔽資料比率較高時, 資料沒有辦法提供充分的訊息, 但貝氏方法配合可靠的先驗分佈, 所得結果會優於最大概似方法。

In a competing risks model, the component fails if any of the risk factors fails. These factors are all from the same component, hence they may be correlated. In this thesis, we consider the competing risks model under bivariate Marshall-Olkin exponential distribution under hybrid censoring which is the mixture of conventional Type I and Type II censoring schemes and is quite useful in lifetesting or reliability experiments. It is often to include masked data in which the risk factor that causes failure of the component is not observed. We apply the maximum likelihood approach via EM algorithm along with the missing information principle to estimate the standard errors of the MLE. Bayesian approach incorporated with subjective prior and noninformative prior is also considered with the aid of MCMC method. Statistical
inference on the model parameters as well as the mean lifetimes and the reliability functions of the component and risk factors is derived. Simulation study shows that the maximum likelihood approach performs poorly when the proportion of the masking data is high due to insufficient information, while Bayesian method can be provide good results with reliable prior information.

[1] Basu, A.P. and Ghosh, J.K. (1978). “Identifiability of the multinormal and other distri-
bution under competing risks model.” J. Multivariate Analysis, 8, 413–429.
[2] Berger, J.O. (1985). Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag.
[3] Berger, J.O. and Sun, D. (1993). “Bayesian analysis for the poly-Weibull distribu-
tion.” Journal of the American Statistical Association, 88, 1412–1418.
[4] Berkson, J. and Elveback, L. (1960). “Competing exponential risks with particular in-
ference to the study of smoking lung cancer.” Journal of the American Statistical Association, 87, 84–89.
[5] Boland, P.J. (1998). “An arrangement increasing property of the Marshall-Olkin bivariate exponential.” Statistics and Probability Letters, 37, 167–170.
[6] Chen, S.M. and Bhattacharya, G.K. (1988). “Exact confidence bounds for an expo-
nential parameter hybrid censoring.” Communications in Statistics-Theory and Methods
, 17, 1858–1870.
[7] Childs, A., Chandrasekhar, B., Balakrishnan, N., and Kundu, D. (2003). “Exact likeli-
hood inference based on type-I and type-II hybrid censored samples from the exponential distribution.” Annals of the Institute of Statistical Mathematics, 55, 319–330.
[9] Dempster, A., Laird, N., and Rubin, D. (1977). “Maximum likelihood from incomplete data via the EM algorithm.” Journal of the Royal Statistical Society, Series B, 39, 1-38.
[10] Draper, N. and Guttman, T. (1987). “Bayesian analysis of hybrid life-test with expo-
nential failure times.” Annals of the Institute of Statistical Mathematics, 39, 219–255.
[11] Efron, B. (1967). “The two sample problem with censored data.” Proceedings of the
Fifth Berkeley Symposium Mathematical Statistics Probability, 4, 831–853.
[12] Efron, B. (1979). “Bootstrap method: another look at the jacknife.” Annals of Statistics, 17, 1-26.
[13] Epstein B. (1954). “Truncated life-tests in the exponential case.” Annals ofMathematical Statistics, 25, 555–564.
[14] Fan, T.H. and Hsu T.M. (2011). “Statistical Inference of a Series System with Masked
Data with Correlated Log-normal Lifetime Distributions under Type-I Censoring. ” International
Conference on Advances in Probability and Statistics - Theory and Applications,
Hong Kong, December 30, 2011.
[15] Geman, S. and Geman, D. (1984). “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images.” IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
[16] Guan Q. and Tang Y. (2012). “Optimal step-stress test under type-I censoring for multivariate exponential distribution.” Journal of Statistical Planning and Inference
, 142, 1908–1923.
[17] jefferys, H. (1961). Theory of Probability. New York: Oxford University Press.
[18] Joe, H. (1990). “Families of Min-Stable multivariate exponential and Multivariate Extreme Value Distributions.” Statistics and Probability Letters, 9, 75–81.
[19] Kaplan, E.L. and Meier, P. (1958). “Non-parametric estimation from incomplete obser-
vation.” Journal of the American Statistical Association, 53, 457–481.
[20] Li, G.A. (2005). “A characterization of the multivariate Marshall-Olkin exponential dis-
tribution and its parameter estimation.” Chinese Journal of Engineering mathematics
, 22, 1055–1062.
[21] Lin, D.K.J., Usher, J.S., and Guess, F.M. (1996). “Bayes estimation of component reliability from masked system-life data.” IEEE Transactions on Reliability, 45, 233–237.
[22] Louis, T.A. (1982). “Finding the observed information matrix using the EM algo-
rithm.” Journal of the Royal Statistical Society, Series B, 44, 226–233.
[23] Ma, Z.S. and Krings, A.W. (2008). “The Competing Risks Analysis Approach to Reliabil-
ity Survivability, and Prognostics and Health Management. ” The 2008 IEEE Aerospace Conference. Big Sky, Montana, March 1-8, 2008.
[24] Marshall, A.W. and Olkin, I. (1967). “A multivariate exponential distribution.” Journal
of the American Statistical Association, 62, 30–44.
[25] Mclachlan G.J. and Krishnan T. (2008). The EM Algorithm and Extensions, Second
Edition. New York: Wiley.
[26] Miyakawa, M. (1984). “Analysis of incomplete data in competing risk model.” IEEE Transactions on Reliability, 33, 293–296.
[27] Sarhan, A.M. (2001). “Reliability estimations of components from masked system life
data.” Reliability Engineering and System Safety, 74, 107–113.
[28] Sarhan, A.M. (2003). “Estimation of system components reliabilities using masked
data.” Applied Mathematics and Computation, 136, 79–92.
[29] Sarkar, S.K. (1987). “A continuous bivariate exponential distribution.” Journal of the
American Statistical Association, 82, 667–675.
[30] Usher, J.S. and Hodgson, T.J. (1988). “Maximum likelihood analysis of component reliability using masked system life-test data.” IEEE Transactions on Reliability, 37, 550-555.