本論文針對不同逐步設限資料(progressively censored data)探討在 log-gamma 和線性失敗率 (linear failure rate) 模式下的估計問題, 以及在韋伯(Weibull)和對數常態(log-normal)模式下的壽命檢測計劃。在形狀(shape)參數已知的情況下, 我們分別以牛頓法, EM 演算法和修正的 EM 演算法來計算逐步型 II 設限資料(progressively Type-II censored data)下log-gamma 分配之位置(location) 和尺度(scale)參數的最大概似估計值, 並利用條件法和蒙地卡羅法求取位置和尺度參數, 百分位數與可靠度函數的信賴區間。我們又分別利用傳統的貝氏推導方式和馬可夫鏈蒙地卡羅(Markov Chain Monte Carlo)法來求取廣義逐步型 II 設限資料(general progressively Type-II censored data)下線性失敗率分配參數的貝氏估計值和其預測值, 並使用不同先驗(prior)分配來探討這些估計值的敏感性分析(sensitivity analysis)。最後, 我們利用模擬退火演算法(simulated annealing algorithm)找出逐步區間設限計劃 (progressively interval censoring plan) 下雙參數韋伯和對數常態模式的最佳檢測時間(optimally spaced inspection times), 並比較四種不同最佳檢測時間所求得的參數最大概似估計值之漸近相對效率 (asymptotic relative efficiency)。

In this dissertation, we first discuss the estimations of parameters for log-gamma and linear failure rate distributions based on different kinds of progressively censored data.  By assuming the shape parameter to be known, we apply three different methods -- Newton-Raphson method, the EM algorithm, and a new modified EM algorithm, to compute the maximum likelihood estimates of location and scale parameters of the log-gamma distribution based on progressively Type-II censored data. We also construct the conditional and unconditional confidence intervals for the location and scale parameters, the quantiles and the reliability function. Next, we employ the conventional Bayesian derivation and the Markov Chain Monte Carlo method to obtain the Bayesian estimates of parameters and predict the missing values and the future samples for the linear failure rate distribution based on the general progressively Type-II censored data. The sensitivity of these estimates to the modest changes in the prior is further examined. We then apply simulated annealing algorithm to determine the optimally spaced inspection times for the Weibull and log-normal distributions for any given progressive interval censoring plan. The comparison of the asymptotic relative efficiencies of the maximum likelihood estimates of the parameters under four different inspection schemes is made at the end.

1 緒論 ...............................................................................................................  1
 1.1 研究動機 ..................................................................................................  1
 1.2 文獻回顧與研究目的................................................................................  2
 1.3 本文架構....................................................................................................  5

2 最大概似估計法 .........................................................................................   7
 2.1 最大概似估計值......................................................................................   9
  2.1.1 最大概似逼近法 (Approximate Maximum Likelihood Estimation).  10
  2.1.2 EM (Expectation-Maximization) 演算法............................................  15
  2.1.3 修正的 EM 演算法...........................................................................  19
 2.2 數值分析..................................................................................................  20

3 區間估計 .....................................................................................................  24
 3.1 最大概似估計值的大樣本性質..............................................................  24
 3.2 蒙地卡羅法..............................................................................................  25
 3.3 條件法 (Conditional Method).................................................................  31
  3.3.1 Z1 和 Z2 的聯合條件分配...........................................................  35
  3.3.2 Z2 的邊際條件分配及參數 σ 的精確信賴區間....................  36
  3.3.3 Z1 的邊際條件累積分配函數及參數 μ 的精確信賴區間........................  37
  3.3.4 百分位數的精確信賴區間和可靠度函數的精確信賴下界............  38
 3.4 數值分析.................................................................................................  39
  3.4.1 模擬比較............................................................................................  39
  3.4.2 範例....................................................................................................  42

4 貝氏估計法 ................................................................................................  44
 4.1 傳統貝氏推論.........................................................................................  45
  4.1.1 貝氏估計............................................................................................  45
  4.1.2 貝氏預測............................................................................................  47
 4.2 馬可夫鏈蒙地卡羅(Markov Chain Monte Carlo)法............................  49
 4.3 最大概似估計法.....................................................................................  54
 4.4 數值分析.................................................................................................  60
  4.4.1 模擬比較............................................................................................  61
  4.4.2 敏感性分析 (Sensitivity Analysis)....................................................  63
  4.4.3 實際資料分析....................................................................................  65

5 逐步型 I 區間設限抽樣下的壽命檢測計劃 ...........................................  70
 5.1 逐步型 I 設限資料下的壽命檢測計劃................................................  71
 5.2 韋伯分配.................................................................................................  74
  5.2.1 費雪情報矩陣....................................................................................  74
  5.2.2 最佳檢測時間與檢測次數................................................................  76
  5.2.3 敏感性分析........................................................................................  79
  5.2.4 檢測方式的比較................................................................................  84
 5.3 對數常態分配.........................................................................................  84
  5.3.1 費雪情報矩陣....................................................................................  88
  5.3.2 最佳檢測時間與檢測次數................................................................  89
  5.3.3 常態分配的對稱性............................................................................  90
  5.3.4 檢測方式的比較................................................................................  91
 5.4 最佳逐步設限計劃 (Optimal Progressive Censoring Scheme).............  91
  5.4.1 韋伯分配............................................................................................  93
  5.4.2 常態分配............................................................................................  94
 5.5 可靠度抽樣計劃 (Reliability Sampling Planning)................................  96
  5.5.1 韋伯分配............................................................................................  97
  5.5.2 對數常態分配..................................................................................  100

6 結論 ..........................................................................................................  102

附錄 A: 模擬退火演算法 ..........................................................................  104
附錄 B: 有關第 5 章對數常態分配的部分模擬結果 ............................  107

參考資料 .....................................................................................................  112



表目錄
2.1 最大概似估計值和最大概似逼近估計值的比較.................................... 14
2.2 資料 A-D 在 log-gamma 分配假設下參數的估計值........................... 22
2.3 不同演算方式所需的疊代次數 ............................................................ 22

3.1 統計量 P1, P2 和 P3 的覆蓋率............................................................  26
3.2 Z1 和 Z2 的模擬百分位數...................................................................... 28
3.3 檢驗 Z1 和 Z2 的模擬百分位數之穩定性的結果.................................. 31
3.4 蒙地卡羅法和條件法的比較: 100 次機率值 P(Pi,2.5% ≦ Zi ≦ Pi,97.5%| a),i=1,2, 的平均......40
3.5 資料 A-D 在 log-gamma 分配假設下, 參數的最大概似估計值及其信賴區間.............. 43
3.6 資料 A-D 在 log-gamma 分配假設下, 百分位數和可靠度函數的最大概似估計值及其信賴下界..............43

4.1 參數 λ 和 υ 在樣本數 n=25 的估計值及標準誤............................ 61
4.2 在先驗分配 α1=1, α2=2.5, β1=1.5, β2=1.5 假設下, 參數 λ 和 υ 在樣本數 n=25 的貝氏估計值及標準誤........................... 62
4.3 參數 λ 和 υ 以及 G1 和 G2 的敏感性分析............................ 64
4.4 洪水量資料在線性失敗率分配假設下的參數貝氏估計值和預測值.............. 67

5.1 韋伯分配在二種不同逐步設限計劃下, 以 ξi=( ti/b)c, i=1,…,m 型式來表示的最佳檢測時間及參數的漸近相對效率, ARE(b) 和 ARE(c), 的值..............77
5.2 表 5.1 在使用準則 CR1 下, 參數 b 和 c 之漸近相對效率的加權組合:          λARE(b) + (1-λ) ARE(c), λ=0.1(0.1)0.9 ..............78
5.3檢測終止時間固定在韋伯分配(b=1,c=1) 之 90% 百分位數和 70% 百分位數情況下的最檢測時間及參數的漸近相對效率 ..............79
5.4 捨入誤差對韋伯分配之參數的漸近相對效率影響的比較 .............................80
5.5 韋伯分配參數 b 和 c 之漸近相對效率的敏感性分析.................................81
5.6 在逐步設限計劃為 q1=q2=…=qm-1=0 和 qm=1 下, 韋伯分配選用不同檢測時間其參數漸近相對效率的比較.............. 85
5.7 在逐步設限計劃為 q1=0.1, q2= 0, q3=0.2 q4=…=qm-1=0 和 qm=1下, 韋伯分配選用不同檢測時間其參數漸近相對效率的比較.............. 86
5.8 在準則 CR1 和逐步設限計劃q1=q2=…=qm-1=0 下, 利用 (5.12) 式的設定方式所得到的最佳檢測時間以及參數的漸近相對效率.............. 91
5.9 在不同逐步設限計劃下, 常態分配選用不同檢測時間其參數漸近相對效率的比較......... 92
5.10 在逐步設限計劃為 q1=…=qm-1=0 的情況下, 常態分配在不同檢測次數以 (t-μ)/σ 型式表示的最佳等距檢測時間..............92
5.11 在 n=200 和 m=8 時, 韋伯分配在選用不同檢測時間和不同期望移除比例的最佳逐步設限計劃 ..............94
5.12 在檢測次數 m=8 時, 韋伯分配之最佳逐步設限計劃的敏感性分析............95
5.13 在 n=200 和 m=5 時, 常態分配在選用不同檢測時間和不同期望移除比例的最佳逐步設限計劃.........95
5.14 在檢測次數 m=5 時, 常態分配之最佳逐步設限計劃的敏感性分析............96
5.15 韋伯分配的可靠度抽樣計劃 .................................................................. 100
5.16 對數常態分配的可靠度抽樣計劃................................................................ 101

6.1 常態分配在二種不同逐步區間設限計劃下, 以 ξi *=(ti-μ)/σ,  i=1,…,m 型式來表示的最佳檢測時間及參數的漸近相對效率, ARE(μ) 和 ARE(σ) 的值..............107
6.2 表 6.1 在使用準則 CR1 下, 參數 μ 和 σ 漸近相對效率的加權組合:           λARE(μ) + (1-λ)ARE(σ), λ=0.1(0.1)0.9 ..............108
6.3 捨入誤差對常態分配之參數的漸近相對效率影響的比較... .........................109
6.4 常態分配參數 μ 和 σ 之漸近相對效率的敏感性分析 ................................... 110



圖目錄
2.1 EM 演算法中參數 μ 和 σ 收斂過程的軌跡圖.........................................20
2.2 資料 A 在 k=10.6204 時, 利用 EM 演算法和修正 EM 演算法參數收斂過程的軌跡圖....23
2.3 資料 B 在 k=8.4504 時, 利用 EM 演算法和修正 EM 演算法參數收斂過程的軌跡圖.....23
4.1 後驗分配機率密度函數在選用不同先驗分配之參數 (α1, α2, β1, β2) 的比較..............66
4.2 利用 Gibbs 抽樣所產生參數 λ 和 υ 的 5000 筆樣本之後驗密度函數的核估計圖形, 樣本的時間序列圖, 以及時差自相關的圖形..............68
4.3 利用 Gibbs 抽樣所產生 (a)Y{1:5}[23] (b) Y{2:5}[23] (c) Y{3:5}[23] (d) Y{4:5}[23] (e) Y{5:5}[23] (f) G1=  Y{j:5}[23] 的 5000 筆樣本之後驗密度函數的核估計圖形, 樣本的時間序列圖, 以及時差自相關的圖形 .............. 68
4.4 G1 後驗密度函數之核估計圖形在不同逐步設限計劃下的比較.............. 69

Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover Publications, New York.

Ashour, S. K. and Youssef, A. (1991). Bayesian estimation of a linear failure rate. Journal of the Indian Association for Productivity, Quality and Reliability, 16, 9-16.

Bain, L. J. and Engelhardt, M. (1973). Interval estimation for the two-parameter double exponential distribution. Technometrics, 15, 875-887.

Bain, L. J. (1974). Analysis for the linear failure-rate life-testing distribution.  Technometrics, 16, 551-559.

Bain, L. J. (1978). Statistical Analysis of Reliability and Life-Testing Models - Theory and Practice. Marcel Dekker, New York.

Balakrishnan, N. (1989). Approximate maximum likelihood estimation of the mean and standard deviation of the normal distribution based on type-II censored samples.  Journal of Statistical Computation and Simulation, 32, 137-148.

Balakrishnan, N. (1990). Approximate maximum likelihood estimation for a generalized logistic distribution. Journal of Statistical Planning and Inference, 26, 221-236.

Balakrishnan, N. and Aggarwala, R. (2000). Progressive Censoring: Theory, Methods, and Applications. Birkhäuser, Boston.

Balakrishnan, N. and Chan, P. S. (1995a). Maximum likelihood estimation for the log-gamma distribution under Type-II censored samples and associated inference. In Advances in Reliability (Ed., N. Balakrishnan), pp.409-437. CRC Press, Boca Raton, Florida.

Balakrishnan, N. and Chan, P. S. (1995b). Maximum likelihood estimation for the three-parameter log-gamma distribution under Type-II censoring. In Advances in Reliability (Ed., N. Balakrishnan), pp.439-451. CRC Press, Boca Raton, Florida.

Balakrishnan, N. and Chan, P. S. (1996). A comparison of conditional and unconditional inference relating to log-gamma distribution. In Lifetime Data: Models in Reliability and Survival Analysis (Eds., N. P. Jewell, A. C. Kimber, M.-L. T. Lee, and G. A. Whitmore), pp.29-37. Kluwer Academic Publishers, Netherlands.
 
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. and Malik, H. J. (1986). Order statistics from the linear exponential distribution, Part I: Increasing hazard rate case. Communications in Statistics-Theory and Methods, 15, 179-203.

Balakrishnan, N. and Sandhu, R. A. (1995). A simple simulational algorithm for generating progressive Type-II censored samples. The American Statistician, 49, 229-230.

Balasooriya, U., Saw, S. L. C. and Gadag V. (2000). Progressively censored reliability sampling plans for the Weibull distribution. Technometrics, 42, 160-167.

Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis (2nd ed.). Springer-Verlag, New York.

Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, Massachusetts.

Chen, G. and Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology, 27, 154-161.

Childs, A. and Balakrishnan, N. (2000). Conditional inference procedures for the Laplace distribution when the observed samples are progressively censored.  Metrika, 52, 253-265.

Cohen, A. C. (1963). Progressively censored samples in life testing. Technometrics, 5, 327-339.

Corana, A., Marchesi, M., Martini, C. and Ridella, S. (1987). Minimizing multi-modal functions of continuous variables with the simulated annealing algorithm. ACM Transactions on Mathematical Software, 13, 262-280.

Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society,  Series B, 39, 1-38.

DiCiccio, T. J. (1987). Approximate inference for the generalized gamma distribution. Technometrics, 29, 33-40.

Fisher, R. A. (1934). Two new properties of mathematical likelihood. Proceedings of the Royal Society, Series A, 144, 285-307.

Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1996). Markov Chain Monte Carlo in Practice. Chapman & Hall, London.

Gumbel, E. J. and Mustafi, C. K. (1967). Some analytical properties of bivariate extremal distributions. Journal of the American Statistical Association, 62, 569-588.

Hager, H. W. and Bain, L. J. (1970). Inference procedures for the generalized gamma distribution. Journal of the American Statistical Association, 65, 1601-1609.

Kantam, R. R. L., Roa, A. V. and Roa, G. S. (2005). Optimum group limits for estimation in scaled log-logistic distribution from a grouped data. Statistical Papers, 46, 359-377.

Kirkpatrick, S., Gelatt, C. D. Jr. and Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220, 671-680.

Kulldorff, G. (1961). Contributions to the Theory of Estimation from Grouped and Partially Grouped Samples. John Wiley & Sons, New York.

Lawless, J. F. (1980). Inference in the generalized gamma and log-gamma distributions. Technometrics, 22, 409-419.

Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data. John Wiley & Sons, New York.
 
Lieberman, G. J. and Resnikoff, G. J. (1955). Sampling plans for inspection by variables. Journal of the American Statistical Association, 50, 457-516.

Lieblein, J. and Zelen, M. (1956). Statistical investigation of the fatigue life of deep groove ball bearings. Journal of Research of the National Bureau of Standards, 57, 273-316.

Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21, 1087-1090.

Nelson, W. (1982). Applied Life Data Analysis. John Wiley & Sons, New York.

Ng, H. K. T., Chan, P. S. and Balakrishnan, N. (2002). Estimation of parameters from progressively censored data using EM algorithm. Computational Statistics & Data Analysis, 39, 371-386.

Ng, H. K. T., Chan, P. S. and Balakrishnan, N. (2004). Optimal progressive censoring plans for the Weibull distribution. Technometrics, 46, 470-481.

O'Neill, B. and Wells, W. T. (1972). Some recent results in lognormal parameter estimation using grouped and ungrouped data. Journal of the American Statistical Association, 67, 76-80.

Pandey, A., Singh, A. and Zimmer, W. J. (1993). Bayes estimation of the linear hazard-rate model. IEEE Transactions on Reliability, 42, 636-640.

Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika, 61, 539-544. 
 
Rao, A. V., Rao, A. V. D. and Narasimham, V. L. (1994). Asymptotically optimal grouping for maximum likelihood estimation of Weibull parameters. Communications in Statistics - Simulation and Computation, 23, 1077-1096. 

Rosaiah, K., Kantam, R. R. L. and Narasimham, V. L. (1991). Optimum class limits for ML estimation in two-parameter gamma distribution from a grouped data. Communications in Statistics - Simulation and Computation, 20, 1173-1189.

Schneider, H. (1989). Failure-censored variables-sampling plans for lognormal and Weibull distributions. Technometrics, 31, 199-206.

Schader, M. and Schmid, F. (1986). Optimal grouping of data from some skew distributions. Computational Statistics Quarterly, 3, 151-159.

Sen, A. (2006). Linear hazard rate distribution. In Encyclopedia of Statistical Sciences (2nd ed.), (Eds., S. Kotz, N. Balakrishnan, C. Read and B. Vidakovic), pp. 4212-4217. John Wiley & Sons, Hoboken, New Jersey.

Sen, A. and Bhattacharyya, G. K. (1995). Inference procedures for the linear failure rate model. Journal of Statistical Planning and Inference, 46, 59-76.

Smith, B. (2005). Bayesian Output Analysis Program (BOA). Program package over the Internet, http://www.public-health.uiowa.edu/boa.

Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics, 33, 1187-1192.

Viveros, R. and Balakrishnan, N. (1994). Interval estimation of parameters of life from progressively censored data. Technometrics, 36, 84-91.