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研究生： 蔡宗憲
Tsai, Tsung-Hsien
論文名稱： 左截斷與右設限資料的分位數餘命函數之信賴帶
Confidence Bands for Quantile Residual Life Function with Left Truncated and Right Censored Data
指導教授： 嵇允嬋
Chi, Yunchan
學位類別： 博士
Doctor
系所名稱： 管理學院 - 統計學系
Department of Statistics
論文出版年： 2015
畢業學年度： 103
語文別： 英文
論文頁數： 60
中文關鍵詞： 分位數餘命函數 、信賴帶 、左截斷與右設限資料 、平賭理論
外文關鍵詞： quantile residual life function, confidence band, left truncated and right censored data, martingale theorem
相關次數： 點閱：98  下載：3 
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    • 使用分位數餘命函數對產品與人體壽命做預測，已廣泛應用在工業製造、臨床醫學、壽險精算等領域。Csörgő and Csörgő (1987) 以及 Chung(1989) 分別在完整資料與右設限資料下，研究分位數餘命的無母數估計以及其大樣本理論性質
    。然而目前文獻上尚未有針對左截斷與右設限資料的大樣本估計理論，因此本論文使用計數過程與平賭理論推導得到樣本分位數餘命函數的弱收歛性。
    近年來，Jeong et al. (2008)、 Zhou and Jeong (2011) 和 Chi et al. (2015) 研究了固定時間點下，中位數餘命的逐點信賴區間。然而，此逐點信賴區間對於中位數餘命函數並無法達到預設立的信心水準，因此本論文使用標準化的分位數餘命函數建構分位數餘命函數的漸近信賴帶。此漸近信賴帶包含樣本分位數的變異數，以及在模擬試驗下具有較正確的信心水準。

    The use of quantile residual life function to make prediction is widely used in many fields, such as reliability, medical studies, actuarial science and business. Csörgő and Csörgő (1987) and Chung(1989) developed the non-parametric large sample estimation theory of quantile (percentile) residual life function with complete data and right censored data respectively. However, the asymptotic theorem for left truncated and right censored is uninvestigated in literature. Therefore, the weak convergence of sample quantile residual life function is derived based on counting process and martingale theorem in this dissertation.
    More recently, Jeong et al. (2008), Zhou and Jeong (2011) and Chi et al. (2015) studied point-wise confidence intervals for median residual lifetime at a particular time point. However, for median residual life function, the point-wise confidence interval generally does not attain the nominal confidence level. Therefore, we proposed a confidence band with corrected confidence level and it is based on normalized quantile residual life process which involves the estimated variance of sample quantile residual life function.

    Contents Chapter 1 Introduction 1 1.1 Background 1 1.2 A motivated dataset 1 1.3 Literature review 2 Chapter 2 Confidence Bands for Quantile Residual Life Function with Right Censored Data 5 2.1 The estimation of quantile residual life function 6 2.1.1 Quantile residual life function 6 2.1.2 Notation 6 2.1.3 Sample quantile residual life function 7 2.2 Weak convergence of sample quantile residual life function 8 2.3 Confidence bands for quantile residual life function 12 2.3.1 Confidence band with estimation of density function 13 2.3.2 Confidence band without estimation of density function 15 2.3.3 Normalized confidence band without estimation of density function 16 Chapter 3 Confidence Bands for Quantile Residual Life Function with Left-Truncated and Right-Censored Data 18 3.1 The estimation of quantile residual life function with LTRC data 18 3.1.1 Notation 18 3.1.2 Sample quantile residual life function with LTRC data 19 3.2 Weak convergence of sample quantile residual life function with LTRC data 21 3.3 Confidence bands for quantile residual life function with LTRC data 24 3.3.1 Confidence band with estimation of density function 24 3.3.2 Confidence band without estimation of density function 26 3.3.3 Normalized confidence band without estimation of density function 27 Chapter 4 Numerical Results 29 4.1 Simulation setting 29 4.2 Comparison of results 30 Chapter 5 An Example 37 5.1 Definition of the social participation group 37 5.2 Confidence bands for median residual life function with social participation 37 Chapter 6 Conclusion and Future Research 40 6.1 Conclusion 40 6.2 Future research 40 Reference 42 Appendix A The Proofs of Lemmas in Chapter 2 44 A.1 Proof of Lemma 2.1 44 A.2 Proof of Lemma 2.2 45 A.3 Proof of Lemma 2.3 46 Appendix B Length Based Estimators for Variance of Sample Quantile Qesidual Function 49 B.1 Length based estimator of 49 B.2 Length based estimator of 50 Appendix C. The Proofs of Lemmas in Chapter 3 52 C.1 Proof of Lemma 3.1 52 C.2 Proof of Lemma 3.2 53 Appendix D Numerical Simulation Results 56 Appendix E Figures for Chapter 5 59

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