System No. U0002-0602201023222600 對極端母體的同時推論和在異質性下選出所有好母體的程序之研究 A study on the simultaneous confidence intervals for all distances from the extreme populations and the procedure of selecting all good populations under heteroscedasticity 淡江大學 管理科學研究所博士班 Graduate Institute of Management Science 98 1 99 余玉如 Yuh-Ru Yu 894560605 博士 English 2010-01-08 156page advisor - Shu-Fei Wu co-chair - Jong-Wuu Wu co-chair - Yi-Ting Hwang co-chair - Wen-Chuan Lee co-chair - Chih-Li Wang co-chair - Liang-Yu Ouyang co-chair - Chin-Chuan Wu 多重型II設限 單階段程序 正確選擇機率 同時推論 子集選擇 雙階段程序 Multiply type II censoring One-stage procedure Probability of correct selection Simultaneous inference Subset selection Two-stage procedure `本論文之內容主要包含「對極端母體的同時推論」和「在異質性下選出所有好母體的程序」兩個主題。在第一個研究主題中，我們考慮k個獨立雙參數指數母體，具有未知的位置參數與共同且未知的尺度參數，在多重型II設限樣本下，利用14種估計量建立14個位置參數與極端母體距離及位置參數與最高極端母體距離之同時信賴區間，並且使用蒙地卡羅模擬法來模擬出臨界值。以最小信賴區間長度為衡量區間估計量表現好壞的準則，考慮在不同的設限組合之下，我們從14種同時信賴區間中選出最好的。文中也提出同時選擇極端母體之子集選擇程序，並舉出兩個數值例子作為極端母體同時推論與子集選擇程序之示範。在第二個研究主題中，考慮k個獨立常態母體，當母體變異數未知且可能不相等時，我們提出設計導向的雙階段程序選出所有好母體，並且證明正確選擇的機率P能夠高出原先設定的機率值P*。然而這種雙階段抽樣程序在第二階段時所需要的額外樣本，有可能因為預算的限制、計畫被終止或是其他的原因而導致無法取得，使得在做統計分析時，只有一組樣本可用，因此我們也提出資料分析的單階段程序選出所有好母體，並且舉一個實例來說明雙階段程序與單階段程序方法之應用。` `This thesis focuses on two topics: the simultaneous confidence intervals (SCIs) for all distances from the extreme populations (the lower extreme population (LEP) and the upper extreme population (UEP)) and the procedure of selecting all good populations under heteroscedasticity. Firstly, 14 SCIs for all distances from the extreme populations and from the UEP for k independent two-parameter exponential populations with unknown location parameters and common unknown scale parameter based on the multiply type II censored samples are proposed. The critical values are obtained by the Monte-Carlo method. The optimal SCIs among 14 methods are identified based on the criteria of minimum confidence length for various censoring schemes. The subset selection procedures of extreme populations are also proposed and two numerical examples are given for illustration. Secondly, suppose that k independent normal populations with means Mu_1,Mu_2,...,Mu_k and variances Sigma-square_1,Sigma-square_2,...,Sigma-square_k are considered. When variances are unknown and possibly unequal, a design-oriented two-stage procedure selecting all good populations such that the probability of correct selection P being greater than a pre-specified value of P* is proposed. When the additional samples at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, a data-analysis one-stage procedure selecting all good populations is proposed. One real-life example is given to illustrate all procedures.` ```Contents Abstract in Chinese.....I Abstract in English.....II Contents.....III List of Tables.....V List of Figures.....VIII List of Notations.....IX Chapter 1 Introduction.....1 1.1 The SCIs for all distances from the extreme populations for two-parameter exponential populations.....1 1.2 A procedure of selecting all good populations for normal populations.....3 1.3 Organization of this dissertation.....4 Chapter 2 The SCIs for all distances from the extreme populations for two-parameter exponential populations.....6 2.1 The SCIs for all distances from the LEP and UEP.....7 2.2 Simulation comparisons.....15 2.3 Subset selection of extreme populations.....18 2.4 Numerical example.....19 2.5 Conclusions.....33 Chapter 3 A procedure of selecting all good populations for normal populations.....34 3.1 Two-stage procedure.....34 3.2 One-stage procedure.....39 3.3 Controlling both types of errors.....42 3.4 Numerical example.....44 3.5 Simulation study.....51 3.6 Conclusions.....56 Chapter 4 Conclusions.....57 4.1 Conclusions for the SCIs for all distances from the extreme populations.....57 4.2 Conclusions for the procedure of selecting all good populations under heteroscedasticity.....58 References.....59 Appendix A Tables.....61 Appendix B Figures.....133 List of Tables Table 2.2.1 The recommended estimators with shorter confidence length than other estimators under various cases for two-sided SCIs.....16 Table 2.2.2 The recommended estimators with shorter confidence length than other estimators under various cases for one-sided SCIs.....17 Table 2.4.1 Simulated data for four populations.....20 Table 2.4.2 The required statistics.....21 Table 2.4.3 The 90% simultaneous confidence intervals.....22 Table 2.4.4 The 95% simultaneous confidence intervals.....23 Table 2.4.5 The 99% simultaneous confidence intervals.....24 Table 2.4.6 Results of subset selection for 1-Alpha=0.90.....25 Table 2.4.7 Results of subset selection for 1-Alpha=0.95.....25 Table 2.4.8 Results of subset selection for 1-Alpha=0.99.....26 Table 2.4.9 Time intervals between failures.....27 Table 2.4.10 The required statistics.....28 Table 2.4.11 The 90% simultaneous confidence intervals.....29 Table 2.4.12 The 95% simultaneous confidence intervals.....30 Table 2.4.13 The 99% simultaneous confidence intervals.....31 Table 2.4.14 Results of subset selection for 1-Alpha=0.90.....32 Table 2.4.15 Results of subset selection for 1-Alpha=0.95.....32 Table 2.4.16 Results of subset selection for 1-Alpha=0.99.....32 Table 3.1.1 Percentage points q*.....37 Table 3.4.1 Bacterial killing ability data.....44 Table 3.4.2 Immediate statistics of four solvents by the initial sample of size n0=14.....45 Table 3.4.3 Immediate statistics of four solvents for the two-stage joint confidence interval.....45 Table 3.4.4 Subset of selected good populations for the two-stage procedure.....45 Table 3.4.5 The 90% and 95% simultaneous inference of Mu_-Mu_i, i=1,2,3,4, for the two-stage procedure.....46 Table 3.4.6 The 90% and 95% simultaneous inference of the ranked parameters for the two-stage procedure.....47 Table 3.4.7 Immediate statistics of four solvents for the single-stage joint confidence interval.....48 Table 3.4.8 Subset of selected good populations for the one-stage procedure.....48 Table 3.4.9 The 90% and 95% simultaneous inference of Mu_-Mu_i, i=1,2,3,4, for the one-stage procedure.....49 Table 3.4.10 The 90% and 95% simultaneous inference of the ranked parameters for the one-stage procedure.....50 Table 3.5.1 The simulation results for P*=0.90.....54 Table 3.5.2 The simulation results for P*=0.95.....55 Table A.1 The critical values of dt for the 90% SCIs for all distances from the LEP and UEP under equal censoring schemes.....61 Table A.2 The critical values of dt for the 95% SCIs for all distances from the LEP and UEP under equal censoring schemes.....66 Table A.3 The critical values of dt for the 99% SCIs for all distances from the LEP and UEP under equal censoring schemes.....71 Table A.4 The critical values of dt for the 90% SCIs for all distances from the UEP under equal censoring schemes.....76 Table A.5 The critical values of dt for the 95% SCIs for all distances from the UEP under equal censoring schemes.....81 Table A.6 The critical values of dt for the 99% SCIs for all distances from the UEP under equal censoring schemes.....86 Table A.7 The critical values of dt for the 90% SCIs for all distances from the LEP and UEP under unequal censoring schemes when k=4.....91 Table A.8 The critical values of dt for the 95% SCIs for all distances from the LEP and UEP under unequal censoring schemes when k=4.....92 Table A.9 The critical values of dt for the 99% SCIs for all distances from the LEP and UEP under unequal censoring schemes when k=4.....93 Table A.10 The critical values of dt for the 90% SCIs for all distances from the UEP under unequal censoring schemes when k=4.....94 Table A.11 The critical values of dt for the 95% SCIs for all distances from the UEP under unequal censoring schemes when k=4.....95 Table A.12 The critical values of dt for the 99% SCIs for all distances from the UEP under unequal censoring schemes when k=4.....96 Table A.13 The values of dt*Theta_t_hat_star/n for the 90% SCIs for all distances from the LEP and UEP under equal censoring schemes.....97 Table A.14 The values of dt*Theta_t_hat_star/n for the 95% SCIs for all distances from the LEP and UEP under equal censoring schemes.....102 Table A.15 The values of dt*Theta_t_hat_star/n for the 99% SCIs for all distances from the LEP and UEP under equal censoring schemes.....107 Table A.16 The values of dt*Theta_t_hat_star/n for the 90% SCIs for all distances from the UEP under equal censoring schemes.....112 Table A.17 The values of dt*Theta_t_hat_star/n for the 95% SCIs for all distances from the UEP under equal censoring schemes.....117 Table A.18 The values of dt*Theta_t_hat_star/n for the 99% SCIs for all distances from the UEP under equal censoring schemes.....122 Table A.19 The values of dt*Theta_t_hat_star/n for the 90% SCIs for all distances from the LEP and UEP under unequal censoring schemes when k=4.....127 Table A.20 The values of dt*Theta_t_hat_star/n for the 95% SCIs for all distances from the LEP and UEP under unequal censoring schemes when k=4.....128 Table A.21 The values of dt*Theta_t_hat_star/n for the 99% SCIs for all distances from the LEP and UEP under unequal censoring schemes when k=4.....129 Table A.22 The values of dt*Theta_t_hat_star/n for the 90% SCIs for all distances from the UEP under unequal censoring schemes when k=4.....130 Table A.23 The values of dt*Theta_t_hat_star/n for the 95% SCIs for all distances from the UEP under unequal censoring schemes when k=4.....131 Table A.24 The values of dt*Theta_t_hat_star/n for the 99% SCIs for all distances from the UEP under unequal censoring schemes when k=4.....132 List of Figures Figure 1.1.1 The multiply type II censoring.....2 Figure 3.4.1 The 90% and 95% simultaneous inference of Mu_-Mu_i, i=1,2,3,4, for the two-stage procedure.....46 Figure 3.4.2 The 90% and 95% simultaneous inference of the ranked parameters for the two-stage procedure.....47 Figure 3.4.3 The 90% and 95% simultaneous inference of Mu_-Mu_i, i=1,2,3,4, for the one-stage procedure.....49 Figure 3.4.4 The 90% and 95% simultaneous inference of the ranked parameters for the one-stage procedure.....50 Figure B.1 The critical values of dt for the 90% SCIs for all distances from the LEP and UEP for 14 estimators under various censoring schemes.....133 Figure B.2 The critical values of dt for the 95% SCIs for all distances from the LEP and UEP for 14 estimators under various censoring schemes.....135 Figure B.3 The critical values of dt for the 99% SCIs for all distances from the LEP and UEP for 14 estimators under various censoring schemes.....137 Figure B.4 The critical values of dt for the 90% SCIs for all distances from the UEP for 14 estimators under various censoring schemes.....139 Figure B.5 The critical values of dt for the 95% SCIs for all distances from the UEP for 14 estimators under various censoring schemes.....141 Figure B.6 The critical values of dt for the 99% SCIs for all distances from the UEP for 14 estimators under various censoring schemes.....143 Figure B.7 The values of dt*Theta_t_hat_star/n for the 90% SCIs for all distances from the LEP and UEP for 14 estimators under various censoring schemes.....145 Figure B.8 The values of dt*Theta_t_hat_star/n for the 95% SCIs for all distances from the LEP and UEP for 14 estimators under various censoring schemes.....147 Figure B.9 The values of dt*Theta_t_hat_star/n for the 99% SCIs for all distances from the LEP and UEP for 14 estimators under various censoring schemes.....149 Figure B.10 The values of dt*Theta_t_hat_star/n for the 90% SCIs for all distances from the UEP for 14 estimators under various censoring schemes.....151 Figure B.11 The values of dt*Theta_t_hat_star/n for the 95% SCIs for all distances from the UEP for 14 estimators under various censoring schemes.....153 Figure B.12 The values of dt*Theta_t_hat_star/n for the 99% SCIs for all distances from the UEP for 14 estimators under various censoring schemes.....155``` ```Bain, L. 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