在聯合逐步型II設限策略下，我們使用傳統方法與EM演算法分別討論Birnbaum-Saunders分佈參數的最大概似估計。然後，我們根據完全搜索方法、鄰域搜索方法和修改式鄰域搜索方法，分別對成本最小化準則、A-最優準則和 D-最優準則決定 Birnbaum-Saunders 分佈之單目標準則最佳化聯合逐步型 II 設限策略。此外，我們根據以上三個準則兩兩配對，決定Birnbaum-Saunders 分佈之複合準則最佳化聯合逐步型 II 設限策略，以及兩個競爭模型，Birnbaum-Saunders分佈和Inverse Gaussian分佈，在複合準則下合理高效的聯合逐步型 II 設限策略。和完全搜索方法比較後，我們發現修改式鄰域搜索方法比鄰域搜索方法較為準確穩定。最後，我們將所使用的方法應用於骨水泥(bone cement)硬度資料，分別對以上三個準則下，討論在Birnbaum-Saunders分佈和Inverse Gaussian分佈下單目標準則最佳化聯合逐步型 II 設限策略和複合準則最佳化聯合逐步型 II 設限策略。

In this paper, we first derive the maximum likelihood estimates of the parameters of Birnbaum and Saunders (BISA) distributions based on the joint progressive Type-II censored (JPC) samples. We also discuss using the EM algorithm to obtain the MLEs of the model parameters. Then, we determine the single-objective criterion optimal JPC schemes for BISA distributions based on the cost minimization criterion, A-optimality criterion, and D-optimality criterion by the complete search method, variable neighborhood search (VNS) algorithm and modified VNS algorithm. These algorithms are compared in terms of accuracy and computation time. In addition, we determine the compound-criterion optimal JPC schemes based on different optimal criteria and the reasonably efficient compound optimal JPC schemes for two competing statistical models, such as the inverse-Gaussian and BISA models. The advantages of using the compound optimal scheme over the single-objective-criterion optimal scheme are demonstrated through a real-life data set on the micro-indentation test about the hardness of polymeric bone cement.

Contents
1 Introduction 1
2 Joint Progressive Type-II Censoring 4
3 Maximum Likelihood Estimation 7
3.1 Likelihood equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 EM algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Numerical Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Optimal JPC Schemes 18
4.1 Single-objective-criterion optimal design . . . . . . . . . . . . . . . . . . . . 18
4.2 Compound optimal design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.1 Different optimal criteria . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.2 Different competing models . . . . . . . . . . . . . . . . . . . . . . . 26
5 Practical Example 31
6 Concluding Remarks 35
Appendices 38
References 54

List of Tables
1 Comparison of simulated expected lengths of life tests based on RBJPC and JPC schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Simulated biases and MSEs of estimates, the sum of four MSEs (TMSE), the computation time (in seconds) (Time) obtained from different methods for the BISA Lifetime distributions with (μ1, μ2) = (0.4, 0.5) and (0.5, 0.6), and (λ1, λ2) = (0.6, 0.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Simulated biases and MSEs of estimates, the sum of four MSEs (TMSE), the computation time (in seconds) (Time) obtained from different methods for the BISA Lifetime distributions with (μ1, μ2) = (1.0, 1.1) and (2.0, 2.1), and (λ1, λ2) = (0.6, 0.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Simulated biases and MSEs of estimates, the sum of four MSEs (TMSE), the computation time (in seconds) (Time) obtained from different methods for the BISA Lifetime distributions with (μ1, μ2) = (0.4, 0.5) and (0.5, 0.6), and (λ1, λ2) = (0.6, 0.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Simulated biases and MSEs of estimates, the sum of four MSEs (TMSE), the computation time (in seconds) (Time) obtained from different methods for the BISA Lifetime distributions with (μ1, μ2) = (1.0, 1.1) and (2.0, 2.1), and (λ1, λ2) = (0.6, 0.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 Optimal JPC schemes obtained through the complete search method, VNS, and MVNS algorithms with different initial inputs R0 for the BISA distributions with small (m, n, k) when μ1 = μ2 = 0.5, λ1 = 0.6 and λ2 = 0.7 and (C1, C2, C3) = (10, 50, 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7 Optimal JPC schemes and corresponding costs under the BISA distributions with different values of (μ1, μ2) and λ1 = 1 and λ2 = 1.5 for given (m, n) and k when (C1, C2, C3) = (10, 50, 250). . . . . . . . . . . . . . . . . . . . . . . . 20
8 Optimal JPC schemes and corresponding costs for the BISA distributions with different values of (μ1, μ2) and (λ1, λ2) when (m, n) = (12, 12) and (C1, C2, C3) = (10, 50, 250) and (20, 50, 250). . . . . . . . . . . . . . . . . . . 22
9 φ1- and φ2-optimal JPC schemes and compound optimal JPC scheme at α∗ for settings (i) and (ii) when (m, n, k) = (12, 12, 5), μ1 = 0.5, μ2 = 0.6, λ1 = 0.6, and λ2 = 0.7, and (C1, C2, C3) = (20, 150, 10). . . . . . . . . . . . . . . . . . 24
10 Compound optimal JPC schemes and their relative efficiencies for the BISA distributions with respect to the parameters μ = (μ1, μ2). . . . . . . . . . . 25
11 Optimal JPC schemes under the BISA and IG models obtained by CSM, VNS, and MVNS algorithms with different initial inputs R0 and (m, n, k) when μ1 = 0.5, μ2 = 0.6, λ1 = 0.6, and λ2 = 0.7 and (C1, C2, C3) = (10, 50, 250). . 28
12 Relative efficiency when the true distribution is IG, but the optimal scheme obtained from the BISA model is used. . . . . . . . . . . . . . . . . . . . . 28
13 φBISA- and φIG-optimal JPC schemes and compound optimal JPC scheme at α∗ for settings (I)–(III) when (m, n, k) = (12, 12, 5), μ1 = β1 = 0.5, μ2 = β2 = 0.6, λ1 = γ1 = 0.6, and λ2 = γ2 = 0.7, and (C1, C2, C3) = (5, 50, 250). . . . . 29
14 The hardness for the treatments 0.1% NH2 and 0.2% NH2 in bone cement specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
15 Goodness-of-fit tests for the indicated treatments under the BISA models for the data set in Table 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
16 Test for common shape parameter under the BISA models for the data set in Table 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
17 φ1- and φ2-optimal JPC schemes and compound optimal JPC scheme at α∗ for settings (i) and (ii) when (m, n, k) = (20, 20, 17), (μ1, μ2, λ1, λ2) = (0.0228, 0.0238, 178.9279, 181.5502), and (C1, C2, C3) = (20, 150, 10). . . . . 33
18 φBISA- and φIG-optimal JPC schemes and compound optimal JPC scheme at α∗ for settings (I) and (II) when (m, n, k) = (20, 20, 17), μ1 = 0.0228, μ2 = 0.0238, λ1 = 178.9279, and λ2 = 181.5502, β1 = 343230.4784, β2 = 321091.9170, γ1 = 178.9745, and γ2 = 181.6015. . . . . . . . . . . . . . . . . 35

List of Figures
1 Illustration of the JPC scheme with the kth failure comes from the X-sample. 5
2 The efficiency plot of φ1- and φ2-efficiencies of Rα versus α ∈ [0, 1] for setting (i), where φ1(R) = det[V(Θ)] and φ2(R) = tr[V(Θ)]. . . . . . . . . . . . . 24
3 The efficiency plot of φ1- and φ2-efficiencies of Rα versus α ∈ [0, 1], for setting (ii) with φ1(R) = tr[V(Θ)] and φ2(R) = C1k + C2E(Wk) + C3tr[V(Θ)]. . 25
4 The efficient plot of φBISA- and φIG-efficiencies of Rα versus α ∈ [0, 1] based on setting (I) (i.e., D-optimality). . . . . . . . . . . . . . . . . . . . . . . . 30
5 The efficient plot of φBISA- and φIG-efficiencies of Rα versus α ∈ [0, 1] based on setting (II) (i.e., A-optimality). . . . . . . . . . . . . . . . . . . . . . . . 30
6 The efficient plot of φBISA- and φIG-efficiencies of Rα versus α ∈ [0, 1] based on setting (III) (i.e., cost minimization criterion). . . . . . . . . . . . . . . . 30
7 The efficiency plot of φ1- and φ2-efficiencies of Rα versus α ∈ [0, 1] for setting (i), where φ1(R) = det[V(Θ)] and φ2(R) = tr[V(Θ)]. . . . . . . . . . . . . 34
8 The efficiency plot of φ1- and φ2-efficiencies of Rα versus α ∈ [0, 1], for setting (ii) with φ1(R) = tr[V(Θ)] and φ2(R) = C1k + C2E(Wk) + C3tr[V(Θ)]. . 34
9 The efficient plot of φBISA- and φIG-efficiencies of Rα versus α ∈ [0, 1] based on setting (I) (i.e., D-optimality). . . . . . . . . . . . . . . . . . . . . . . . 36
10 The efficient plot of φBISA- and φIG-efficiencies of Rα versus α ∈ [0, 1] based on setting (II) (i.e., A-optimality). . . . . . . . . . . . . . . . . . . . . . . . 36

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