在可靠度分析裡，加速壽命試驗是藉由將物件置於較正常環境嚴厲的應力水準下，來減少試驗時間的一種方法；而階段加速壽命試驗則是將同一批物件置於逐漸增加的應力水準下的加速試驗。不同的設限計畫也常被應用在加速壽命試驗，而在壽命試驗中資料又可分為紀錄各物件失效時間的完整資料與只觀測失效的個數群組資料兩種。本文討論資料來自壽命分配為指數分配、單一應力、群集資料下的逐步型I 設限階段加速壽命試驗之最大概似與貝氏推論。由於在可靠度試驗中通常樣本數較少，使得傳統上基於大樣本漸近結果的最大概似推論不甚準確，但貝氏方法卻可在小樣本時提供較穩定的估計。另外，將根據 V-準則、D-準則與 A-準則，分別決定最佳的試驗時間與最佳的階段應力增量。

Accelarated life test (ALT) is a widely used technique to reduce the experiment time. Experimenters abridge the time by placing the items at the more severe stress levels than at nomal-use contidion. Besides, in the step-stress ALT, the stress levels are gradually increasing. Also, censoring scheme is often applied when the life test is to be executed. Complete data, which record all failure times and grouped data, which only include the numbers of failure items are two types of data in life test. In this thesis, assume that the lifetime of each item follows an exponential
distribution under the single stress progressive Type-I censoring ALT with grouped data. Maximum likelihood as well as Bayesian inferences on the related parameters are developed. The traditional MLE according as the large sample properties is not precise enough since there is no large sample in reliability test. However Bayesian method would offer stable estimation when the sample size is not large. Futhermore, the search for optimal experiment time and optimal stress increment is derived, which is based on V-optimality, D-optiamlity and A-optimality.

[1] Bai, D.S., and Kim, M.S. (1993) Optimum simple step-stress accelerated life
tests for weibull distribution and type I censoring. Naval Research Logistics 40,
193-210.
[2] Bai, D.S., Kim, M.S. and Lee, S.H. (1989) Optimum simple step-stress accelerated
life tests with censoring. IEEE Transactions on Reliability 38, 528-532.
[3] Balakrishnan, N. and Aggarwala, R. (2000) Progressive Censoring: Theory,
Methods, and Applications. Birkhauser, Boston.
[4] Balakrishnan, N. and Han, D. (2008) Optimal step-stress testing for progressively
Type-I censored data from exponential distribution. J. Statist. Plann.
Inference to appear.
[5] Balakrishnan, N. and Sandhu, R.A. (1996) Best linear unbiased and maximum
likelihood estimation for exponential distribution under progressive type-II censored
smaples. Sankhya 58, 1-9.
[6] Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer, New York.
[7] Casella, G. and Berger, R.L. (2002) Statistical Inference, 2nd edn. Duxbury,
Pacific Grove, CA.
[8] Chib, S. and Greenberg, E. (1995) Understanding the Metropolis-Hastings algorithm.
Amer. Statist. 49, 327-335
[9] Cohen, A.C. (1963) Progressively censored samples in life testing. Technometries
5, 327-329.
[10] Cohen, A.C. and Norgaard, N.J. (1977) Progressively censored sampling in the
three-parameter gamma distribution. Technometrics 19, 333-340.
[11] Drop, J.R. and Mazzuchi, T.A. (2004) A general Bayes exponential inference
model for accelerated life test. J Stat Plan Inference 119, 55-74.
[12] Fan T.H., Wang W.L. and Balakrishnan, N. (2008) Exponential progressive
step-stress life-testing with link function based on Box-Cox transformation.
Journal of Statistical Planning and Inference 138, 2340-2354.
[13] Gibbons, D. I. and Vance, L. C. (1983) Estimators for the 2-parameter Weibull
distribution with progressively censored samples. IEEE Transactions on Reliability
32, 95-99.
[14] Gouno, E., Sen, A. and Balakrishnan, N. (2004) Optimal step-stress test under
progressive Type-I censoring. IEEE Transactions on Reliability 53, 383-393.
[15] Khamis, I.H. (1997) Optimum M-step, step-stress test with k stress variables.
Comm. Statist. Comput. Simul. 26, 1301-1313.
[16] Hastings, W.K. (1970) Monte Carlo sampling method using Makov chain and
their applications. Biometrika 57, 97-109.
[17] Jeffreys, H. (1961) Theorey of Probability, 3rd edn. Oxford University Press,
London.
[18] Khamis, I.H. and Higgins, J.J. (1998) A new model for step-stress testing. IEEE
Transactions on Reliability 47, 131-134.
[19] Mann, N.R. (1971) Best linear invariant estimation for weibull parameter under
pregressive censoring. Technometrics 13, 521-533.
[20] Meeker, W.Q. and Hann, G.J. (1985) How to plan an accelerated life test.
American Socirty for Quality Control Statistics Division 10.
[21] Miller, R. and Nelson, W. (1983) Optimum simple step stress plans for accelerated
life testing. IEEE Transactions on Reliability 32, 59-65.
[22] Nelson,W. (1980) Accelerated life testing - step-stress models and data analysis.
IEEE Transactions on Reliability 29, 103-108.
[23] Nelson, W. (1990) Accelerated Testing: Statistical Models, Test Plans, and Data
Analyses. John Wiley & Sons, New York.
[24] Robert, C.P. (2001) The Bayesian Choice : From Decision-Theoretic Foundations
to Computational Implementation. Springer, New York.
[25] Tang, L.C., Sun, Y.S., Goh, T.N. and Ong, H.L. (1996) Analysis of step-stress
accelerated-life-test data: a new approach. IEEE Transactions on Reliability
51, 69-74.
[26] Tang, L.C., Sun, Y.S., Goh, T.N. and Ong, H.L. (1999) Planning accelerated
life tests for censored two-parameter exponential distributions. Naval Research
Logistics 16, 169-186.
[27] Teng, S.L. and Yeo, K.P. (2002) A least-squares approach to analyzing lifestress
relationship instep-stress accelerated life tests. IEEE Transactions on
Reliability 51, 177-182.
[28] Thisted, R.A. (1988) Elements of Statistical Computing: NUMERICAL COMPUTATION
(Elements of Statistical Computing). Chapman & Hall/CRC.
[29] Wu, S.J. and Chang, C.T. (2003) Inference in the Pareto distribution based
on progressive Type II censoring with random removals. J. Appl. Statist 30,
163-172.
[30] Wu, S.J., Lin, Y.P. and Chen, Y.J. (2006) Planning step-stress life test with
progressively type I group-censored exponential data. Neerlandica 60, 46-56.
[31] Wong, J. S. (1993) Simultaneously estimating the three Weibull parameters
from progressively censored samples. Microelectronics and Reliability 33, 2217-
2224.
[32] Xiong, C.(1998) Inferences on a simple step-stress model with type-II censored
exponential data. IEEE Transactions on Reliability 47, 142-146.
[33] Xiong, C. and Ji, M. (2004) Analysis of grouped and censored data from stepstress
life test. IEEE Transactions on Reliability 53, 22-28.
[34] Yuen, H.K. and Tse, S.K. (1996) Parameters estimation for weibull distributed
lifetimes under progressive censoring with random removeals. Journal of Statistical
Computation and Simulation 57-71.