多個非負隨機變數的強韌概似法

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姓名

唐翊維(Yi-Wei Tang) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

多個非負隨機變數的強韌概似法
(Robust likelihood inference for comparing several populations of nonnegative random variables)

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摘要(中)

本文提出利用強韌概似函數 (robust likelihood function) 方法比較多個非負隨機變數。此強韌檢定法是將多個獨立的伽瑪概似函數強韌化，得到強韌華德檢定統計量、強韌分數檢定統計量及強韌概似比檢定統計量，亦利用強韌檢定法建立共同平均數之信賴區間。利用模擬與實例分析，先比較強韌檢定法、Krishnamoorthy and Oral (2015) 提出的標準化概似比檢定統計量(standardized likelihood ratio test) 和 Bebu and Methew (2007) 利用廣義樞紐量(generalized pivot statistic) 建構之廣義信賴區間 (generalized confidence interval)，再比較利用強韌檢定法建立共同平均數之信賴區間和 Krishnamoorthy and Oral (2015) 改良 Zou et al. (2009) 提出的 MOVER 近似信賴區間方法建立之共同平均數信賴區間。

摘要(英)

In this thesis, we propose a robust likelihood function method to analyze several nonnegative random variables. The robustness test method is not only for robustizing several independent gamma likelihood function to obtain robust Wald-type test statistic, robust score test statistic, and robust likelihood ratio test statistic, but also for constructing the confidence interval of common mean. By using both simulation and real data analysis, we can demonstrate the merit of our technique of robustness. At first, we compare the differences among robustness test method, the standardized likelihood ratio test proposed by Krishnamoorthy and Oral (2015), and the generalized confidence interval constructed by Bebu and Mathew (2007) which used the generalized pivot statistic. Additionally, we also compare the differences between the common mean confidence interval established by the MOVER approximate confidence interval method proposed by Krishnamoorthy and Oral (2015) modified Zou et al. (2009).

關鍵字(中)

★ 成對設計
★ 強韌概似函數
★ 強韌分數檢定
★ 廣義信賴區間

關鍵字(英)

★ Paired design
★ Robust likelihood function
★ Robust score test
★ Generalized confidence interval

論文目次

目錄
摘要................................................................................................................................. i
Abstract .......................................................................................................................... ii
致謝辭.......................................................................................................................... iii
目錄............................................................................................................................... iv
表目錄........................................................................................................................ viii
圖目錄......................................................................................................................... xvi
第一章緒論.................................................................................................................. 1
第二章文獻回顧.......................................................................................................... 2
2.1 標準化概似比檢定............................................................................................. 2
2.2 廣義變數近似法................................................................................................. 2
2.2.1 廣義 P 值 ..................................................................................................... 3
2.2.2 廣義樞紐量.................................................................................................. 3
2.2.3 廣義信賴區間 ............................................................................................ 4
2.2.4 多組獨立對數常態分配平均數之廣義變數近似法.................................. 4
2.2.5 二元對數常態分配平均數之廣義變數近似法.......................................... 5
2.2.6 共同平均數之廣義變數近似法.................................................................. 7
2.3 變異數估計恢復法信賴區間............................................................................. 7
第三章三組獨立伽瑪模型之強韌化........................................................................ 10
v
3.1 伽瑪實作模型可被強韌化................................................................................ 11
3.2 強韌化伽瑪實作模型........................................................................................ 13
3.2.1 矩陣 I ......................................................................................................... 13
3.2.2 矩陣V ......................................................................................................... 16
3.3 實作模型下之修正項 A 和 B ............................................................................ 25
3.3.1 感興趣參數為 2 3  , 下之修正項 A 和 B ................................................... 25
3.3.2 感興趣參數為單一參數下之修正項 A 和 B ............................................. 28
3.4 強韌變異數估計量............................................................................................ 29
3.4.1 感興趣參數為 2 3  , 下之強韌變異數估計量........................................... 29
3.4.2 感興趣參數為單一參數下之強韌變異數估計量..................................... 30
3.5 強韌華德檢定統計量........................................................................................ 31
3.5.1 感興趣參數為 2 3  , 下之強韌華德檢定統計量....................................... 31
3.5.2 感興趣參數為單一參數下之強韌華德檢定統計量................................. 32
3.6 強韌分數檢定統計量........................................................................................ 32
3.6.1 感興趣參數為 2 3  , 下之強韌分數檢定統計量....................................... 32
3.6.2 感興趣參數為單一參數下之強韌分數檢定統計量................................. 36
3.7 強韌概似比檢定統計量.................................................................................... 39
3.8 共同平均數之強韌推論.................................................................................... 41
3.8.1 共同平均數之修正項................................................................................. 41
vi
3.8.2 共同平均數之強韌變異數估計量............................................................. 42
3.8.3 共同平均數之強韌華德檢定統計量......................................................... 42
3.8.4 共同平均數之強韌分數檢定統計量......................................................... 43
3.8.5 共同平均數之強韌概似比檢定統計量..................................................... 43
第四章多組獨立伽瑪模型之強韌化........................................................................ 45
4.1 伽瑪實作模型可被強韌化................................................................................ 46
4.2 強韌化伽瑪實作模型........................................................................................ 48
4.2.1 矩陣 I ......................................................................................................... 48
4.2.2 矩陣V ......................................................................................................... 51
4.3 實作模型下之修正項 A 和 B ............................................................................ 58
4.3.1 感興趣參數為多個參數下之修正項 A 和 B ............................................. 58
4.3.2 感興趣參數為單一參數下之修正項 A 和 B ............................................. 61
4.4 強韌變異數估計量............................................................................................ 63
4.4.1 感興趣參數為多個參數下之強韌變異數估計量..................................... 63
4.4.2 感興趣參數為單一參數下之強韌變異數估計量..................................... 64
4.5 強韌華德檢定統計量........................................................................................ 65
4.5.1 感興趣參數為多個參數下之強韌華德檢定統計量................................. 65
4.5.2 感興趣參數為單一參數下之強韌華德檢定統計量................................. 65
4.6 強韌分數檢定統計量........................................................................................ 66
vii
4.6.1 感興趣參數為多個參數下之強韌分數檢定統計量................................. 66
4.6.2 感興趣參數為單一參數下之強韌分數檢定統計量................................. 69
4.7 強韌概似比檢定統計量.................................................................................... 73
第五章模擬研究........................................................................................................ 75
5.1 對數常態分配之平均數估計............................................................................ 75
5.2 對數常態分配平均數比例之變異數估計........................................................ 76
5.3 資料生成方式.................................................................................................... 86
5.4 模擬結果............................................................................................................ 89
第六章實例分析...................................................................................................... 189
6.1 實例一.............................................................................................................. 189
6.2 實例二.............................................................................................................. 194
6.3 實例三.............................................................................................................. 198
6.4 實例四.............................................................................................................. 202
第七章結論.............................................................................................................. 205
參考文獻.................................................................................................................... 206
附錄............................................................................................................................ 208

參考文獻

Bebu, I. and Mathew, T. (2008). Comparing the means and variances of a bivariate log-normal distribution. Statistics in Medicine, 27: 2684-2696.
Krishnamoorthy, K. and Oral, E. (2015). Standardized likelihood ratio test for comparing several log-normal means and confidence interval for the common mean. Statistical Methods in Medical Research, 26: 2919–2937.
Li, X. (2009). A generalized p-value approach for comparing the means of several log-normal populations. Statistics and Probability Letters, 79: 1404-1408.
Royall, R. M. and Tsou, T. S. (2003). Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions. Journal of the Royal Statistical Society, Series B, 65: 391-104.
Silva, E. L., and Lisboa, P. (2007). Analysis of the characteristic features of the density functions for gamma, Weibull and log-normal distributions through RBF network pruning with QLP. World Scientific and Engineering Academy and Society, 6: 223-228.
Tian, L. and Wu, J. (2007). Inferences on the common mean of several log-normal populations: the generalized variable approach. Biometrical Journal, 49: 944-951.
Tsou, T. S. and Chen, C. H. (2008). Comparing several means of dependent populations of count-A parametric robust approach. Statistics in Medicine, 27: 2576-2585.
Tsui, K.-W. and Weerahandi, S. (1989). Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. Journal of the American Statistical Association, 84: 602–607.
Weerahandi, S. (1993). Generalized confidence intervals. Journal of the American Statistical Association, 88: 899-905.
Zou, G. Y., Taleban, J. and Huo, C. Y. (2009). Simple confidence intervals for log-normal means and their differences with environmental applications. Environmentrics, 20: 172-180.
Zou, G. Y. and Donner, A. (2008). Construction of confidence limits about effect measures: a general approach. Statistics in Medicine, 27: 1693-1702.
Zou, G. Y., Huo, C. Y. and Taleban, J. (2009). Confidence interval estimation for log-normal data with application to health economics. Computational Statistics and Data Analysis, 53: 3755-3764.

指導教授

鄒宗山(Tsung-Shan Tsou)

審核日期

2021-7-28

推文