在生存分析中考量到成本的因素時，區間資料經常被使用為產品壽命分析數據。本論文主要是在探討雙參數韋伯分配(Weibull Distribution)在區間設限資料的情況下，透過非參數估計之KM(Kaplan-Meier)法、最大概似估計法(Maximum Likelihood Estimation)及貝氏法(Bayesian Approach)進行評估衡量。本文將採用DE(Differential Evolution)演算法及EM(Expectation-Maximization)演算法來求得最大概似估計值，貝氏法則採用Lindley近似估計法及馬可夫鏈蒙地卡羅法(Markov Chain Monte Carlo)估計其參數。此外，參數之區間估計亦可求得。市場回饋區間資料亦可用來分析及應用。由本研究案例中得到在絕對比例誤差的值使用貝氏法比其它兩種方法較優。

In survival analysis, the inspection costs should be concerned and the interval censored data are often occurred in lifetime data analysis. In this study, we consider the non-parametric approach of Kaplan-Meier (KM) method, the maximum likelihood estimation (MLE), and the Bayesian approach and evaluate their performance of the Weibull distribution with the interval censored data. Here, MLE via the Differential Evolution (DE) algorithm and the Expectation–Maximization (EM) algorithm and Bayesian approach via Lindley’s Approximation and Markov Chain Monte Carlo (MCMC) are used to estimate parameters. Additionally, the confidence intervals for these estimators are obtained. The field return data are used to illustrate the applications. The results show that the Bayesian approach outperforms the MLE and KM methods in term of mean of absolute percentage error in most cases.

[1]Ardia, D., Boudt, K., Carl, P., Mullen, K., and Peterson, B.G., “Differential evolution with DEoptim: An application to non-convex portfolio optimization,’’ The R Journal, 3(1), 27-34 (2011).
[2]Balakrishnan, N., “Progressive censoring methodology: an appraisal,” Test, 16, 211-259 (2007).
[3]Balakrishnan, N. and Kateri, M., “On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data,” Statistics & Probability Letters, 78, 2971-2975 (2008).
[4]Blischke, W.R., Karim, M.R., and Murthy, D.P., Warranty Data Collection and Analysis, Springer Science & Business Media, New York (2011).
[5]Dempster, A.P., Laird, N.M., and Rubin, D.B., “Maximum likelihood from incomplete data via the EM algorithm,” Journal of the Royal Statistical Society. Series B (Methodological), 1-38 (1977).
[6]Flygare, M.E., Austin, J.A., and Buckwalter, R.M., “Maximum likelihood estimation for the 2-parameter Weibull distribution based on interval-data,” IEEE Transactions on Reliability, 34, 57-59 (1985).
[7]Kaplan, E.L. and Meier, P., “Nonparametric estimation from incomplete observations,” Journal of the American Statistical Association, 53, 457-481 (1958).
[8]Keats, J.B., Lawrence, F.P., and Wang, F., “Weibull maximum likelihood parameter estimates with censored data,” Journal of Quality Technology, 29, 105 (1997).
[9]Lindley, D.V., “Approximate bayesian methods,” Trabajos de Estadística y de investigación Operativa, 31, 223-245 (1980).
[10]Martz, H.F. and Waller, R., Bayesian Reliability Analysis, John Wiley & Sons, New York, (1982).
[11]Menon, M., “Estimation of the shape and scale parameters of the Weibull distribution,” Technometrics, 5, 175-182 (1963).
[12]Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E., “Equation of state calculations by fast computing machines,” The Journal of Chemical Physics, 21, 1087-1092 (1953).
[13]Mullen, K., Ardia, D., Gil, D.L., Windover, D., and Cline, J., “DEoptim: An R package for global optimization by differential evolution,” Journal of Statistical Software, 40, 1-26 (2011).
[14]Naikan, V.A., Reliability Engineering and Life Testing, PHI Learning Pvt. Ltd., India (2008).
[15]Nelson, W., Applied Life Data Analysis, John Wiley & Sons, New York (1982).
[16]Ng, H.K.T. and Wang, Z., “Statistical estimation for the parameters of Weibull distribution based on progressively type-I interval censored sample,” Journal of Statistical Computation and Simulation, 79, 145-159 (2009).
[17]Pang, W. K., Hou, S. H., and Yu, W. T., “On a proper way to select population failure distribution and a stochastic optimization method in parameter estimation,” European Journal of Operational Research, 177, 604-611 (2007).
[18]Price, K., Storn, R.M., and Lampinen, J.A., Differential evolution: A Practical Approach to Global Optimization, Springer Science & Business Media, New York (2006).
[19]Singh, R.S. and Totawattage, D.P., “The statistical analysis of interval-censored failure time data with applications,” Open Journal of Statistics, 3, 155-166 (2013).
[20]Sinha, S.K. and Sloan, J., “Bayes estimation of the parameters and reliability function of the 3-parameter Weibull distribution,” IEEE Transactions on Reliability, 37, 364-369 (1988).
[21]Smith, A.F. and Roberts, G.O., “Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods,” Journal of the Royal Statistical Society. Series B (Methodological), 3-23 (1993).
[22]Sttvig, J.G., Censored Weibull Distributed Data in Experimental Design, Master thesis, Norwegian University of Science and Technology, Norwegian (2014).
[23]Storn, R. and Price, K., “Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces,” Journal of Global Optimization, 11, 341-359 (1997).
[24]Sun, J., The Statistical Analysis of Interval-Censored Failure Time Data, Springer Science & Business Media, New York (2007).
[25]Tan, Z., “A new approach to MLE of Weibull distribution with interval data,” Reliability Engineering & System Safety, 94, 394-403 (2009).
[26]Wang, F. K., “Using BBPSO algorithm to estimate the Weibull parameters with censored data,” Communications in Statistics-Simulation and Computation, 43, 2614-2627 (2014).
[27]Wang, F. K. and Cheng, Y., “EM algorithm for estimating the Burr XII parameters with multiple censored data,” Quality and Reliability Engineering International, 26, 615-630 (2010).
[28]Weibull, W., “A statistical distribution function of wide applicability,” Journal of Applied Mechanics, 13, 293-297 (1951).