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系統識別號 U0026-1308201514271700
論文名稱(中文) 左截斷與右設限資料的分位數餘命函數之信賴帶
論文名稱(英文) Confidence Bands for Quantile Residual Life Function with Left Truncated and Right Censored Data
校院名稱 成功大學
系所名稱(中) 統計學系
系所名稱(英) Department of Statistics
學年度 103
學期 2
出版年 104
研究生(中文) 蔡宗憲
研究生(英文) Tsung-Hsien Tsai
電子信箱 maninbluetsai@gmail.com
學號 R28971026
學位類別 博士
語文別 英文
論文頁數 60頁
口試委員 指導教授-嵇允嬋
口試委員-陳珍信
口試委員-陳瑞彬
口試委員-張淑惠
口試委員-張升懋
口試委員-蘇佩芳
中文關鍵字 分位數餘命函數  信賴帶  左截斷與右設限資料  平賭理論 
英文關鍵字 quantile residual life function  confidence band  left truncated and right censored data  martingale theorem 
學科別分類
中文摘要 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。
英文摘要 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.
論文目次 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
參考文獻 Aly, E. E. A. (1997). Nonparametric tests for comparing two mean residual life functions. Lifetime data analysis, 3(4), 353-366.

Chi, Y., Tsai, T. H., Tu, Y. H., and Tsai, W. Y. (2015). Comparison of several confidence intervals for median residual lifetime with left-truncated and right-censored data. Communications in Statistics-Simulation and Computation. Advance online publication. doi: 10.1080/03610918.2013.870199.

Chiao, C. Y., Lee, S. H., Liao, W. C., Yen, C. H., Lin, Y. J., Li, C. R., Lai, T. J., Lin, H. S., Lee, M. S. and Lee, M. C. (2013). Social participation and life expectancy—the case of older adults in Taiwan from 1996 to 2003. International Journal of Gerontology, 7(2), 97-101.

Chukova, S., Arnold, R. and Wang, D. Q. (2004). Warranty analysis: An approach to modeling imperfect repairs. International Journal of Production Economics, 89(1), 57-68.

Chung, C. J. F. (1989). Confidence bands for percentile residual lifetime under random censorship model. Journal of Multivariate Analysis, 29(1), 94-126.

Csörgő, M., and Csörgő, S. (1987). Estimation of percentile residual life.Operations Research, 35(4), 598-606.

Csörgő, S., and Viharos, L. (1992). Confidence bands for percentile residual lifetimes. Journal of statistical planning and inference, 30(3), 327-337.

Fleming, T. R., and Harrington, D. P. (1991). Counting processes and survival analysis. John Wiley & Sons.
Franco-Pereira, A. M., Lillo, R. E., and Romo, J. (2012). Comparing quantile residual life functions by confidence bands. Lifetime data analysis, 18(2), 195-214.

Greitzer, F. L. and Ferryman, T. A. (2001). Predicting remaining life of mechanical systems. In Intelligent Ship Symposium IV (pp. 2-3).

Jeong, J. H. (2014). Statistical inference on residual life. New York: Springer.

Jeong, J. H., Jung, S. H. and Costantino, J. P. (2008) Nonparametric inference on median residual life function. Biometrics, 64, 157–163.

Kaplan, E. L., and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American statistical association, 53(282), 457-481.

Kosorok, M. R. (1999). Two-sample quantile tests under general conditions.Biometrika, 86(4), 909-921.
Lai, T. L., and Ying, Z. (1991). Estimating a distribution function with truncated and censored data. The Annals of Statistics, 417-442.

Parzen, M. I., Wei, L. J., and Ying, Z. (1997). Simultaneous confidence intervals for the difference of two survival functions. Scandinavian Journal of Statistics,24(3), 309-314.

Wang, M. C., Jewell, N. P., Tsai, W. Y. (1986). Asymptotic properties of the product limit estimate under random truncation. The Annals of Statistics, 14, 1597-1605.

Wang, Y., Liu, P., and Zhou, Y. (2014). Quantile residual lifetime for left-truncated and right-censored data. Science China Mathematics, 1-18.

Tsai, T. H., Tsai, W. Y., Chi, Y. C. and Chang, S. M. (2015). “Estimation of the ratio of two median residual lifetimes with left-truncated and right-censored data,” and accept by Biometrics.

Tsai, W. Y., Jewell, N. P., & Wang, M. C. (1987). A note on the product-limit estimator under right censoring and left truncation. Biometrika, 74(4), 883-886.

Tse, S. (2005). Quantile process for left truncated and right censored data. Annals of the Institute of Statistical Mathematics, 57(1), 61-69.

Zhou, M. and Jeong, J. H. (2011). Empirical likelihood ratio test for median and mean residual lifetime. Statistics in Medicine 30, 152-159.
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