本論文延續Hsu（2014）的結果討論對數位置尺度壽命分配在型I混合設限定應力加速壽命試驗的點估計與區間估計。 由於使用最大概似法求得參數，經常不能獲得具體公式求解，因而必須改用數值演算法運算求得。因此，我們提出以近似最大概似估計法所求得的解，作為任何數值演算法的初始值，再進一步求得最大概似估計值。我們特別針對韋伯分配和對數常態分配比較最大概似估計值和近似最大概似估計值的偏差(bias)和均方誤差(mean squared error)。 此外，我們根據最大概似估計值討論四種區間估計:常態近似分配，概似比(likelihood ratio)，和兩個參數跋靴方法(parametric bootstrap methods)。最後，我們用兩個實際例子來說明我們的方法。

In this thesis, we extend the work of Hsu (2014) to discuss the inference on constant stress accelerated life tests terminated by a hybrid Type-I censoring at the first stress level.  The model is based on a general log-location-scale lifetime distribution with mean life which is a linear function of stress, along with constant scale. From the work of Hsu (2014), it is observed that the maximum likelihood estimates (MLEs) of the unknown parameters cannot be obtained in a closed form. We propose the approximate maximum likelihood estimates (AMLEs) and these can be used as initial estimates for any iterative procedure. We then evaluate the bias, and mean square error of these estimators; and provide asymptotic, likelihood ratio, and bootstrap confidence intervals for the parameters of the Weibull and lognormal distributions with the MLE. Finally,the results are illustrated with two examples.

Abstract 1
1 Introduction 2
2 Assumptions and Model Description 6
3 Maximum Lilelihood Estimation 8
4 Approximate Maximum Likelihood Estimation 10
4.1 Log-normal Distribution  10
4.2 Weibull Distrbution  14
5 Confidence Intervals 19
5.1 Approximate Confidence Interval  19
5.2 Likelihood Ratio-Based Confidence Interval  19
5.3 Bootstrap Confidence Intervals 20
5.3.1 Percentile Bootstrap Confidence Interval 21
5.3.2 Bootstrap BCa Percentile Interval 21
6 Simulation Results 22
7 Data Analysis 29
8 Concluding Remarks 32

Bagdonavicius, V. and Nikulin, M. (2002). Accelerated Life Models: Modeling and StatisticalAnalysis. Chapman & Hall/CRC, Boca Raton, Florida.
Banerjee, A. and Kundu, D. (2008). Inference based on Type-II hybrid censored data from a Weibull distribution. IEEE Transactions on Reliability, 57, 369–378.
Balakrishnan, N. (2009). A synthesis of exact inferential results for exponential step-stress models and associated optimal accelerated life-tests. Metrika 69, 351–396.
Balakrishnan, N. and Kundu, D. (2013). Hybrid censoring: models, inferential results and applications.Computational Statistics & Data Analysis 57, 166–209.
Balakrishnan, N., Kannan, N., Lin, C. T. and Ng, H. K. T. (2003). Point and interval estimation for Gaussian distribution based on progressively Type-II censored samples.IEEE Transactions on Reliability, 52, 90–95.
Balakrishnan, N., Kannan, N., Lin, C. T. and Wu, S. J. S. (2004) Inference for the extreme
value distribution under progressive Type-II censoring. Journal of Statistical
Computation and Simulation, 74, 25–45.
Ding, C., Yang, C. and Tse, S. K. (2010). Accelerated life test sampling plans for the Weibull distribution under Type I progressive interval censoring with random removals. Jour-nal of Statistical Computation and Simulation 80, 903–914.
Dube, S., Pradhan, B. and Kundu, D. (2011). Parameter estimation of the hybrid censored log-normal distribution. Journal of Statistical Computation and Simulation 81, 275–287.
Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman &Hall/CRC, Boca Raton, Florida.
Fan, T. H. and Yu, C. H. (2013). Statistical inference on constant stress accelerated life tests under generalized gamma lifetime distributions. Quality and Reliability Engineering International 29, 631–638.
Fan, T. H. and Hsu, T. M. (2014). Constant stress accelerated life test on a multiplecomponent
series system under Weibull lifetime distributions. Communications in Statistics – Theory and Methods 43, 2370–2383.
Gouno, E. and Balakrishnan, N. (2001). Step-stress accelerated life test. In Handbook of Statistics 20: Advances in Reliability (Eds., N. Balakrishnan and C. R. Rao), 623–639, North-Holland, Amsterdam.
Guan, Q., Tang, Y., Fu, J. and Xu, A. (2014). Optimal multiple constant-stress accelerated
life tests for generalized exponential distribution. Communications in Statistics –
Simulation and Computation 43, 1852–1865.
Han, D. and Ng, H. K. T. (2013). Comparison between constant-stress and step-stress
accelerated life rests under time constraint. Naval Research Logistics, 60, 541–556.
Hsu, Y. Y. (2014). On constant stress accelerated life tests terminated by Type-I hybrid
censoring at one of the stress levels. Master Thesis, Tamkang University, Taiwan.
Kundu, D. (2007). On hybrid censored Weibull distribution. Journal of Statistical Planning
and Inference 137, 2127–2142.
Leon, R. V., Ramachandran, R., Ashby, A. J. and Thyagarajan, J. (2007). Bayesian modeling
of accelerated life tests with random effects. Journal of Quality Technology 39, 3–16.
Liu, X. (2012). Planning of accelerated life tests with dependent failure modes based on a gamma frailty model. Technometrics, 54, 398–409.
Liu, X. and Tang, L. C. (2010). Accelerated life test plans for repairable systems with
multiple independent risks. IEEE Transactions on Reliability, 59, 115–127.
Ma, H. and Meeker, W. Q. (2008). Optimum step-stress accelerated life test plans for loglocation-
scale distributions. Naval Research Logistics 55, 551–562.
Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. John
Wiley & Sons, New York.
Nelson, W. B. (1990). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis.
John Wiley & Sons, New York.
Nelson, W. B. (2005a). A bibliography of accelerated test plans. IEEE Transactions on
Reliability 54, 194–197.
Nelson, W. B. (2005b). A bibliography of accelerated test plans, Part II–references. IEEE
Transactions on Reliability 54, 370–373.
Pascual, F. (2008). Accelerated life test planning with independent Weibull competing risks.
IEEE Transactions on Reliability, 57, 435–444.
Shao, J. (2003). Mathematical Statistics. Second edition. Springer, New York.
Tang, L. C. (2003). Multiple-steps step-stress accelerated life test. In Handbook of Reliability
Engineering (Ed., H. Pham), 441–455, Springer, London.
Tiku, M. L., Tan, W. Y. and Balakrishnan, N. (1986). Robust Inference. Marcel Dekker,
New York.
Van Dorp, J. R. and Mazzuchi, T. A. (2005). A general Bayes Weibull inference model for
accelerated life testing. Reliability Engineering and System Safety 90, 140–147.
Wang, B. X., Yu, K. and Sheng, Z. (2014). New inference for constant-stress accelerated life
tests with Weibull distribution and progressively Type-II censoring. IEEE Transac-
tions on Reliability, 63, 807–815.
Watkins, A. J. (1994). Review: Likelihood method for fitting Weibull loglinear models to
accelerated life-test data. IEEE Transactions on Reliability, 43, 361–365.
Watkins, A. J. and John, A. M. (2008). On constant stress accelerated life tests terminated
by Type II censoring at one of the stress levels. Journal of Statistical Planning and
Inference 138, 768–786.
Wiel, S. A. V. and Meeker, W. Q. (1990). Accuracy of approx confidence bounds using
censored Weibull regression data from accelerated life tests. IEEE Transactions on
Reliability 39, 346–351.
Yu, I. T. and Chang, C. L. (2012). Applying Bayesian model averaging for quantile estimation
in accelerated life tests. IEEE Transactions on Reliability 61, 74–83.
Yang, G. B. (1994). Optimum constant-stress accelerated life-test plans. IEEE Transactions
on Reliability 43, 575–581.
Zhang, J., Cheng, G., Chen, X., Han, Y., Zhou, T. and Qiu, Y. (2014). Accelerated life test
of while OLED based on lognormal distribution. Indian Journal of Pure & Applied
Physics 52, 671–677.