系統識別號 U0026-2307201316002500 論文名稱(中文) 隨機右設限資料的核密度估計之帶寬選擇 論文名稱(英文) Bandwidth selection for kernel density estimate for randomly right-censored data 校院名稱 成功大學 系所名稱(中) 統計學系碩博士班 系所名稱(英) Department of Statistics 學年度 101 學期 2 出版年 102 研究生(中文) 游惠群 研究生(英文) Hui-Chun Yu 學號 R28931034 學位類別 博士 語文別 英文 論文頁數 64頁 口試委員 指導教授-吳鐵肩口試委員-任眉眉口試委員-李隆安口試委員-馬瀰嘉口試委員-樊采虹 中文關鍵字 收斂速度  訊息界限  核密度估計  特徵函數  設限資料  帶寬選擇 英文關鍵字 Bandwidth selection  characteristic function  censored data  convergence rate  information bound  kernel density estimation 學科別分類 中文摘要 本文考慮在隨機樣本數為n的隨機右設限(randomly right-censored)資料下，以核密度方法估計(kernel density estimator)存活時間之機率密度函數(lifetime density)f之帶寬(bandwidth)選擇問題。此法將Chiu(1992)提出的帶寬選擇法由完整資料(complete data)推廣至隨機右設限資料。其關鍵在於修正樣本特徵函數(sample characteristic function)的高頻區，使得高頻區降低的變異比增加的偏誤更為顯著。在f與核函數(kernel function)符合特定的平滑條件下，本文提出的帶寬選擇法以最佳(root n)速度收斂至常態分佈，並合理猜測其漸近變異數達到訊息界線(information bound)。模擬研究設定了符合實務的樣本數與設限比率，結果顯示本文提出之帶寬選擇法表現優異，更甚於交叉驗證(cross-validation)選擇法。 英文摘要 Based on randomly right-censored sample of size n, the problem of selecting the global bandwidth in kernel density estimation of lifetime density f is investigated. A stabilized bandwidth selector, which is an extension to censored data of the complete-sample selector of Chiu (1992), is proposed. The key idea of our selector is to modify the weighted sample characteristic function beyond some cut-off frequency to reduce the sample variations without significantly inflating the bias. It is shown that under some smoothness conditions on f and the kernel, our selector is asymptotically normal distributed with the optimal root n relative convergence rate and attains the (conjectured) information bound. The excellent performances of the proposed selector at practical sample sizes are clearly demonstrated in simulation studies. In particular, the proposed selector performs conclusively better than the one selected by cross-validation. 論文目次 1 Introduction 1 2 Literature review 3 2.1 Kernel estimate for complete data case . . . . . . . . . . . . . . . . . . 3 2.2 Boundary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Kernel estimator for right censored data . . . . . . . . . . . . . . . . . 9 3 The proposed method 14 3.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 The Proposed Bandwidth Selector . . . . . . . . . . . . . . . . . . . . . 16 3.3 The Main Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 The Modification of the Proposed Bandwidth Selector . . . . . . . . . . 19 4 Simulation results 21 5 Discussion and future research 23 6 Proofs 24 Appendix A Tables 37 Appendix B Figures 46 Bibliography 61 List of Tables A.1 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A.2 Simulation results for the kernel estimation of Gamma(10,1) . . . . . . 39 A.3 Simulation results for the kernel estimation of Weibull(10,20) . . . . . . 40 A.4 Simulation results for the kernel estimation of Gamma(4,1) . . . . . . . 41 A.5 Simulation results for the kernel estimation of Weibull(4,20) . . . . . . 42 A.6 Simulation results for the kernel estimation of Exponential(1) . . . . . 43 A.7 Simulation results for the kernel estimation of Bimodal I . . . . . . . . 44 A.8 Simulation results for the kernel estimation of Bimodal II . . . . . . . . 45 List of Figures 2.1 The effect of bandwidth on kernel estimate. . . . . . . . . . . . . . . . 4 2.2 Boundary effects on exponential distribution. . . . . . . . . . . . . . . . 8 2.3 (a)An example of the weight function;(b)Pre-weighted exponential distribution with mean equals to 1. . . . . . . . . . . . . . . . . . . . . . . 9 3.1 Cut-off frequency selection. . . . . . . . . . . . . . . . . . . . . . . . . . 20 B.1 Estimated bandwidth density of βSTA, β∞ and βCV for the model ♯ 1 . 47 B.2 Estimated bandwidth density of βSTA, β∞ and βCV for the model ♯ 2 . 48 B.3 Estimated bandwidth density of βSTA, β∞ and βCV for the model ♯ 3 . 49 B.4 Estimated bandwidth density of βSTA, β∞ and βCV for the model ♯ 4 . 50 B.5 Estimated bandwidth density of βSTA, β∞ and βCV for the model ♯ 5 . 51 B.6 Estimated bandwidth density of βSTA, β∞ and βCV for the model ♯ 6 . 52 B.7 Estimated bandwidth density of βSTA, β∞ and βCV for the model ♯ 7 . 53 B.8 Estimated density of the model ♯ 1 . . . . . . . . . . . . . . . . . . . . 54 B.9 Estimated density of the model ♯ 2 . . . . . . . . . . . . . . . . . . . . 55 B.10 Estimated density of the model ♯ 3 . . . . . . . . . . . . . . . . . . . . 56 B.11 Estimated density of the model ♯ 4 . . . . . . . . . . . . . . . . . . . . 57 B.12 Estimated density of the model ♯ 5 . . . . . . . . . . . . . . . . . . . . 58 B.13 Estimated density of the model ♯ 6 . . . . . . . . . . . . . . . . . . . . 59 B.14 Estimated density of the model ♯ 7 . . . . . . . . . . . . . . . . . . . . 60 參考文獻 1. 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