基于右删失宽相依数据的Kaplan-Meier估计和风险率估计的渐近性质
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摘要
本文在生存时间和删失时间均为宽相依数据下,建立了生存函数的Kaplan-Meier估计和风险率估计的强逼近和强表示,获得的强逼近和强表示误差项的收敛速度达到O(n~(-1/2)log~(1/2)n).所得结果推广了负相协和负超可加相依数据情形下的相关结果.
Consider the survival function and hazard rate estimators by the Kaplan-Meier method based on censored data, where the survival and censoring times come from the widely orthant dependent date, respectively. Under some more mild conditions, the uniform strong approximation rates and strong representation for the survival function and hazard rate are investigated, and their uniform strong approximation rates and remainders of strong representation also are obtained with the order O(n~(-1/2) log~(1/2)n) a.s. Our results established generalize the corresponding ones of negatively associated and negatively superadditive dependent data in the related literatures.
Consider the survival function and hazard rate estimators by the Kaplan-Meier method based on censored data, where the survival and censoring times come from the widely orthant dependent date, respectively. Under some more mild conditions, the uniform strong approximation rates and strong representation for the survival function and hazard rate are investigated, and their uniform strong approximation rates and remainders of strong representation also are obtained with the order O(n~(-1/2) log~(1/2)n) a.s. Our results established generalize the corresponding ones of negatively associated and negatively superadditive dependent data in the related literatures.
引文
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[2] Lo S H, Mack Y P, Wang J L. Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator. Probability Theory and Related Fields,1989, 80(3):461-473
[3]周勇,吴长凤.随机左截断数据下乘积限估计的强逼近及其应用.应用数学学报,1999, 22(4):614-620(Zhou Y, Wu C F. A strong approximation of Product-lLimit estimator for random left truncated data with applications. Acta Mathematice applicatae Sinica, 1999, 22(4):614-620)
[4] Liebscher E. Kernel density and hazard rate estimation for censored data underα-mixing condition.Annals of the institute of statistical mathematics, 2002, 54(1):19-28
[5]刘玉涛,史兴杰,周勇.左截断右删失数据下光滑分布函数估计效率.数学学报,2013, 56(5):625-636(Liu Y T, Shi X J, Zhou Y. Efficiency of Smoothed Distribution Function Estimator with Left Truncated and Right Censored Data. Acta Mathematica Sinica, 2013, 56(5):625-636)
[6] Wu Q Y, Chen P Y. Strong representation results of the Kaplan-Meier estimator for censored negatively associated data. Journal of Inequalities and Applications, 2013, 2013(1):1-9
[7] Shi J H, Chen X P, Zhou Y. The strong representation for the nonparametric estimator of length-biased and right-censored data. Statistics and Probability Letters, 2015, 104:49-57
[8] Shen A T, Wang X H. Kaplan-Meier estimator and hazard estimator for censored negatively superadditive dependent data. Statistics, 2016, 50(2):377-388
[9] Cheng F X. Asymptotic properties of hazard rate estimator in censored linear regression. Sankhya:The Indian Journal of Statistics, 2017, 79-A(1):1-12
[10] Wang K Y, Wang Y B, Gao W. Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodology And Computing In Applied Probability,2013, 15(1):109-124
[11] Liu L. Precise large deviations for dependent random variables with heavy tails. Statistics and Probability Letters,2009, 79(9):1290-1298
[12] Joag-Dev K, Proschan F. Negative association of random variables with applications. The Annals of Statistics, 1983, 11(1):286-295
[13] Hu T Z. Negatively superadditive dependence of random variables with applications. Journal of Applied Probability and Statistics, 2000, 16(2):133-144
[14] Wang Y B, Cheng D Y. Basic renewal theorems for random walks with widely dependent increments.Journal of Mathematical Analysis and Applications, 2011, 384(2):597-606
[15] Shen A T. Bernstein-type inquality for widely dependent sequence and its application to nonparametric regression models. Abstract and Applied Analysis, 2013, 2013(1):309-338
[16] Wang X, Xu C, Hu T C, Volodin A, Hu S H. On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST, 2014, 23(3):607-629
[17] Chen W, Wang Y B, Cheng D Y. An inequality of widely dependent random variables and its applications. Lithuanian Mathematical Journal, 2016, 56(1):16-31
[18] Wang X J, Deng X, Hu S H. On consistency of the weighted least squares estimators in a semiparametric regression model. Metrika, 2018, 81(7):797-820
[19] Li Y M, Zhou Y, Liu C. On the convergence rates of kernel estimator and hazard estimator for widely dependent samples. Journal of Inequalities and Applications, 2018, 2018(71):1-10
[20]杨善朝.NA样本最近邻密度估计的相合性.应用数学学报, 2003, 26(3):385-395(Yang S C. Consistency of nearest neighbor estimator of density function for negative associated samples. Acta Mathematice applicatae Sinica, 2003, 26(3):385-395)
[2] Lo S H, Mack Y P, Wang J L. Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator. Probability Theory and Related Fields,1989, 80(3):461-473
[3]周勇,吴长凤.随机左截断数据下乘积限估计的强逼近及其应用.应用数学学报,1999, 22(4):614-620(Zhou Y, Wu C F. A strong approximation of Product-lLimit estimator for random left truncated data with applications. Acta Mathematice applicatae Sinica, 1999, 22(4):614-620)
[4] Liebscher E. Kernel density and hazard rate estimation for censored data underα-mixing condition.Annals of the institute of statistical mathematics, 2002, 54(1):19-28
[5]刘玉涛,史兴杰,周勇.左截断右删失数据下光滑分布函数估计效率.数学学报,2013, 56(5):625-636(Liu Y T, Shi X J, Zhou Y. Efficiency of Smoothed Distribution Function Estimator with Left Truncated and Right Censored Data. Acta Mathematica Sinica, 2013, 56(5):625-636)
[6] Wu Q Y, Chen P Y. Strong representation results of the Kaplan-Meier estimator for censored negatively associated data. Journal of Inequalities and Applications, 2013, 2013(1):1-9
[7] Shi J H, Chen X P, Zhou Y. The strong representation for the nonparametric estimator of length-biased and right-censored data. Statistics and Probability Letters, 2015, 104:49-57
[8] Shen A T, Wang X H. Kaplan-Meier estimator and hazard estimator for censored negatively superadditive dependent data. Statistics, 2016, 50(2):377-388
[9] Cheng F X. Asymptotic properties of hazard rate estimator in censored linear regression. Sankhya:The Indian Journal of Statistics, 2017, 79-A(1):1-12
[10] Wang K Y, Wang Y B, Gao W. Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodology And Computing In Applied Probability,2013, 15(1):109-124
[11] Liu L. Precise large deviations for dependent random variables with heavy tails. Statistics and Probability Letters,2009, 79(9):1290-1298
[12] Joag-Dev K, Proschan F. Negative association of random variables with applications. The Annals of Statistics, 1983, 11(1):286-295
[13] Hu T Z. Negatively superadditive dependence of random variables with applications. Journal of Applied Probability and Statistics, 2000, 16(2):133-144
[14] Wang Y B, Cheng D Y. Basic renewal theorems for random walks with widely dependent increments.Journal of Mathematical Analysis and Applications, 2011, 384(2):597-606
[15] Shen A T. Bernstein-type inquality for widely dependent sequence and its application to nonparametric regression models. Abstract and Applied Analysis, 2013, 2013(1):309-338
[16] Wang X, Xu C, Hu T C, Volodin A, Hu S H. On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST, 2014, 23(3):607-629
[17] Chen W, Wang Y B, Cheng D Y. An inequality of widely dependent random variables and its applications. Lithuanian Mathematical Journal, 2016, 56(1):16-31
[18] Wang X J, Deng X, Hu S H. On consistency of the weighted least squares estimators in a semiparametric regression model. Metrika, 2018, 81(7):797-820
[19] Li Y M, Zhou Y, Liu C. On the convergence rates of kernel estimator and hazard estimator for widely dependent samples. Journal of Inequalities and Applications, 2018, 2018(71):1-10
[20]杨善朝.NA样本最近邻密度估计的相合性.应用数学学报, 2003, 26(3):385-395(Yang S C. Consistency of nearest neighbor estimator of density function for negative associated samples. Acta Mathematice applicatae Sinica, 2003, 26(3):385-395)