本論文針對聯合型II逐步設限策略下，討論韋伯(Weibull)、逆高斯(inverse Gaussian)、和Birnbaum-Saunders分配的參數估計，進而可以估計P(Y<X)。參數估計的求法主要根據三種方法:以牛頓(Newton-Raphson)法解概似方程式，以及隨機EM演算法和雙重退火演算法最佳化概似函數。因為韋伯(Weibull)分配的概似方程式只有一個未知變數，所以也可用二分(Bisection)法來解概似方程式。我們特別針對Weibull分配進行不同參數組合在四種方法下各種設限策略的參數估計模擬研究，並發現使用雙重退火算法所得的最大參數估計值的平均誤差與均方誤差(MSE)的表現是最佳。相同的結論也可在inverse Gaussian和Birnbaum-Saunders分配下得到。最後，我們用所提出的方法來估計一個真實數值例子之P(Y<X)。

In this thesis, the estimation of parameters of a certain family of two-parameter lifetime distributions based on joint progressively Type-II right-censored sample is studied. This family includes the Weibull, inverse Gaussian, and Birnbaum-Saunders distributions. Different numerical methods are used to compute the maximum likelihood estimates of the unknown parameters. These methods include the Newton-Raphson method, stochastic EM algorithm, and dual annealing algorithm with Python. These methods are compared via a Monte Carlo simulation study in terms of their biases and mean squared errors. Recommendations are made from the simulated results and a real data set has been analyzed for illustrative purposes.

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 7
2.3 Dual Annealing Algorithm 9
2.4 Bisection Method Algorithm 11
3 Parameter Estimation for the Weibull Lifetime istributions 12
3.1 Likelihood Equations 12
3.2 The M-step in SEM Algorithm 14
4 Parameter Estimation for the Inverse Gaussian Lifetime Distributions 16
4.1 Likelihood Equations 16
4.2 The M-step in SEM Algorithm 17
5 Parameter Estimation for the Birnbaum-Saunders Lifetime Distributions 19
5.1 Likelihood Equations 19
5.2 The M-step in SEM Algorithm 20
6 Numerical Analysis 22
6.1 Simulation Study 22
6.2 An Illustrative Example 29
7 Concluding Remarks 32
8 Appendix 33
9 References 38

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, λ1 = 0.6, μ2 = 0.6, λ2 = 0.7 . . . . . . . . . . . 23
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 . . . . . . . . . . . 24
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 . . . . . . . . . . . . . . . . . . . . . 25
Table 4 Average values of biases and MSE of the MLE of the parameters in the Weibull distributions with μ = 1, λ1 = 0.5, λ2 = 2 . . . . . . . . . . . . . . . . . . . . . . .26
Table 5 Average values of biases and MSE of the MLE of the parameters in the inverse Gaussian Lifetime distributions with μ1 = 0.5, λ1 = 1.5, μ2 = 0.6, λ2 = 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Table 6 Average values of biases and MSE of the MLE of the parameters in the Birnbaum-Saunders Lifetime distributions with μ1 = 1, λ1 = 0.8, μ2 = 0.5, λ2 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Table 7 Breakdown times of the insulating fluid at two voltage levels, and the corresponding MLE of μ and λ, log-likelihood, KS test statistics, and p values of the fitted Weibull models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Table 8 p values of the LR and KS tests for common shape parameter . . . . . 29
Table 9 The generated JPC samples based on four schemes . . . . . . . . . . . . 30
Table 10 The MLE of Q for different censoring schemes . . . . . . . . . . . . . . . . .31
Table 11 The 95% approximate, Boot-p, and Boot-t confidence intervals and widths for Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

List of Figures
Figure 1 Schematic representation of JPC scheme . . . . . . . . . . . . . . . . . . . . . 2

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