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研究生： 陳佳妏
Chen, Chia-Wen
論文名稱： 極值分配中參數估計之研究
The Study of Parameter Estimation in Extreme Value Distributions
指導教授： 路繼先
Lu, Chi-Hsien Joseph
學位類別： 碩士
Master
系所名稱： 管理學院 - 統計學系
Department of Statistics
論文出版年： 2004
畢業學年度： 92
語文別： 英文
論文頁數： 77
中文關鍵詞： 起始值 、門檻值的決定 、最大概似估計量
外文關鍵詞： Initial value, Maximum Likelihood Estimation, Threshold Detemination
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    • 　　在廣義極值分配最大概似估計量的估計中, 由於起始值的決定並非容易, 於是我們針對目前現有的方法進行探討, 提供一套合理且可行的方法, 建立最大概似估計值搜尋的起始值.
    　　在廣義柏拉圖分配關於門檻值決定的部分, 我們以簡單的方法建立門檻值搜尋的區間.且我們將整個決定門檻值的過程程式化, 自動化.另外我們建立一套明清楚的法則以共最大概似估計量的搜尋.

    　　In the maximum likelihood estimation of the parameters of generalized extreme distribution, the selecting initial values is not a trial task, we study the current available approach and propose an alternative way to establish a reasonable and feasible initial values. As for the determination of the threshold in the generalized Pareto distribution, we propose a simple way to set up the nterval for searching threshold, along with programing the procedure of determining an appropriate threshold and providing a clear-out rule for determining threshold value.

    Contents 1 Introduction 7 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Extreme Value Models 9 2.1 Block maxima model . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Threshold Excess Model . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Max Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Relationship Between GEV and GPD . . . . . . . . . . . . . . . . . . 15 2.5 N-Year Return Level . . . . . . . . . . . . . . . . . . . . . . . . .16 2.5.1 Generalized Extreme Value Distribution . . . . . . . . . . . . . ..17 2.5.2 Generalized Pareto Distribution . . . . . . . . . . . . . . . . . .18 3 Estimation of Generalized Extreme Value Distribution 20 3.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 ML Estimates for Generalized Extreme Value Distribution . . . . . . .21 3.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Searching ML Estimates for Generalized Extreme Value Distribution 23 3.4.1 What Happened to The 10th Dataset . . . . . . . . . . . . . . . . .25 3.4.2 Solve The Problem . . . . . . . . . . . . . . . . . . . . . . . . .26 3.4.3 Further Examine The Change . . . . . . . . . . . . . . . . . . . . 29 3.5 Possible Modication . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Estimation of the Generalized Pareto Distribution 34 4.1 Graphical Approach for the Choice of Threshold . . . . . . . . . . . 35 4.1.1 Mean Residual Life Plot . . . . . . . . . . . . . . . . . . . . . .35 4.1.2 Stability Plot . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Goodness-of-t Tests Approach . . . . . . . . . . . . . . . . . . . .39 4.2.1 Estimation of Parameters . . . . . . . . . . . . . . . . . . . . . 39 4.2.2 Goodness-of-t Tests . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Comment on The Current Approach . . . . . . . . . . . . . . . . . . .41 4.4 Modication and Improvement . . . . . . . . . . . . . . . . . . . . .43 4.4.1 Searching for Maximum Likelihood Estimates . . . . . . . . . .. . .44 4.4.2 Improvement for Calculational Convenience . . . . . . . . . . . . .45 4.4.3 The Modication of Rules for Search . . . . . . . . . . . . . . . .47 4.5 Revisit of Wheaton River data . . . . . . . . . . . . . . . . . . . .49 5 Further Result 54 5.1 Revisit the Simulated Datasets . . . . . . . . . . . . . . . . . . . 54 5.1.1 Initial Value Interval . . . . . . . . . . . . . . . . . . . . . . 54 5.1.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Examine Return Level . . . . . . . . . . . . . . . . . . . . . . . . 56 References 63 Appendix ML Estimates for GEV 65 Appendix ML Estimates for GPD 69 Appendix R functions 73

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