本論文針對韋伯(Weibull)的聯合型 II 逐步設限資料，討論其參數估計和最佳設限策略。其中參數估計使用二分(Bisection)法、牛頓(Newton-Raphson)法、隨機 EM 算法和雙重退火算法最佳化概似函數。最佳設限策略是在給定觀察到的失敗個數、平均的花費時間和共變異矩陣的單位成本下，所尋找到的累計成本最小的設限策略，其中共變異矩陣和成本的關係是以Ａ優化、Ｄ優化作計算。而因共變異矩陣的成本在實務上並不好取得，也另外嘗試限制其他兩個得成本上限，去尋找其中共變異矩陣之Ａ、Ｄ優化最小的。

This thesis aims to estimate parameters and determine the optimal life-testing plans with Weibull distributed lifetimes under joint progressive Type-II censoring scheme proposed by Mondal and Kundu (2019). We obtain the maximum likelihood esti mators numerically by four procedures–the Newton-Raphson and Bisection methods, and stochastic EM and dual annealing algorithms, and determine the optimum joint progressive Type-II censoring schemes by the variable-neighborhood-search-based ap proach discussed in Bhattacharya et al. (2016). A sensitivity analysis is also considered to study the effect of misspecification of parameter values on the optimal scheme.

Contents
1 Introduction 1
2 Likelihood Estimation of Model Parameters 6
2.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Stochastic Expectation-Maximization Algorithm . . . . . . . . . . . . . . . . 8
2.3 Dual Annealing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Bisection Method Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Parameter Estimation for the Weibull Lifetime Distributions 13
3.1 Likelihood Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 The M-Step in SEM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Numerical Analysis 19
4.1 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Optimal JPC Schemes 25
5.1 Procedure for Constructing Neighborhoods . . . . . . . . . . . . . . . . . . . 26
5.2 Algorithm for Obtaining Optimal JPC Scheme . . . . . . . . . . . . . . . . . 26
5.3 Optimal JPC Scheme based on Cost Minimization . . . . . . . . . . . . . . . 28
6 Concluding Remarks 33
7 Appendix 34
8 References 37



List of Tables
Table 1 Average values of biases and MSE of the MLE of the parameters in the
Weibull distributions with µ1 = 0.5, µ2 = 0.6, λ1 = 0.6, λ2 = 0.7. . . . . . . . 20
Table 2 Average values of biases and MSE of the MLE of the parameters in the
Weibull distributions with µ1 = 1.3, λ1 = 0.5, µ2 = 1.1, λ2 = 1.2. . . . . . . . 21
Table 3 Average values of biases and MSE of the MLE of the parameters in the
Weibull distributions with µ = 0.5, λ1 = 0.5, λ2 = 1. . . . . . . . . . . . . . 22
Table 4 Average values of biases and MSE of the MLE of the parameters in the
Weibull distributions with µ = 1.2, λ1 = 0.5, λ2 = 0.8. . . . . . . . . . . . . 23
Table 5 Failure times of air-conditioning systems in two airplanes, and the corre sponding MLE of µ and λ, log-likelihood, KS test statistics, and p values of
the fitted Weibull models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Table 6 MLE, p values of the LR and KS tests for common shape parameter. . 24
Table 7 Optimal JPC schemes obtained through the complete search method
and VNS algorithm with different initial inputs R0 for the Weibull distribu tions with µ = 0.5, λ1 = 0.5, and λ2 = 1 when (m, n, k) = (12, 12, 5) and
(C1, C2, C3) = (10, 50, 250). . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Table 8 Optimal JPC schemes and corresponding costs under the Weibull distri butions with different values of µ and λ1 = 0.5 and λ2 = 1 for given (m, n)
and k when (C1, C2, C3) = (10, 50, 250). . . . . . . . . . . . . . . . . . . . . 29
Table 9 Optimal JPC schemes and their relative efficiencies compared to µ0 = 1
for the Weibull distributions with µ on the case µ1 = µ2, and λ1 = 0.5 and
λ2 = 1 when (m, n) = (12, 12), k = 8 and (C1, C2, C3) = (10, 50, 250). . . . . 30
Table 10 Optimal JPC schemes and their relative efficiencies compared to µ0
for
the Weibull distributions with µ on the case µ1 ̸= µ2, and λ1 = 0.5 and λ2 = 1
when (m, n) = (15, 15), k = 11 and (C1, C2, C3) = (10, 50, 250). . . . . . . . 30
Table 11 Optimal JPC schemes and corresponding costs for the Weibull distribu tions with different values of µ and λ1 = 0.5 and λ2 = 1 when (m, n) = (12, 12)
and (C1, C2, C3) = (10, 50, 250). . . . . . . . . . . . . . . . . . . . . . . . . 31
Table 12 Optimal JPC schemes for the optimization problem in (3) under the
Weibull distributions with µ = 7, λ1 = 6 and λ2 = 5 when (m, n) = (12, 12),
k = 5 and (C1, C2) = (10, 50). . . . . . . . . . . . . . . . . . . . . . . . . . . 32



List of Figures
Figure 1 Illustration of JPC scheme with kth failure comes from Sample X. . . 2
Figure 2 Procedure for obtaining the neighborhood Ni(R0), i = 1, . . . , imax. . . 27
Figure 3 VNS algorithm for obtaining optimal JPC scheme. . . . . . . . . . . . 27

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