删失指标随机缺失下回归函数的复合分位数回归估计
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摘要
在非参数回归模型中,传统的Nadaraya-Watson核估计和局部多项式估计常常因为误差为重尾情况而变得不稳健,Kai等人(2010)提出的复合分位数回归方法能弥补这一缺陷.文章在删失指标随机缺失的情况下,研究了误差具有异方差结构的非参数删失回归模型,利用局部多项式方法构造了回归函数的复合分位数回归估计,并得到了该估计的渐近正态性结果,把Kai等人(2010)的结果推广到删失指标随机缺失的右删失数据下.最后通过模拟发现,尤其是当误差为重尾分布时,该估计方法比Wang和Zheng (2014)提出的核估计方法更好.
In the nonparametric regression model, the Nadaraya-Watson kernel estimation and local polynomial estimation are not robust when the error is heavy tailed. The composite quantile regression method proposed by Kai, et al.(2010)can overcome the shortcoming of robustness. In this paper, we consider the nonparametric regression model with heteroscedastic error when the data are right-censored and the censoring indicators are missing at random, construct the composite quantile regression estimators of regression function based on the local polynomial method,and establish the asymptotic normality of these estimators, which extends the results of the Kai, et al.(2010) to right-censored data with the censoring indicators missing at random. The simulation studies show that our estimators perform better than the kernel estimation proposed by Wang and Zheng(2014), especially when the error is the heavy tail distribution.
In the nonparametric regression model, the Nadaraya-Watson kernel estimation and local polynomial estimation are not robust when the error is heavy tailed. The composite quantile regression method proposed by Kai, et al.(2010)can overcome the shortcoming of robustness. In this paper, we consider the nonparametric regression model with heteroscedastic error when the data are right-censored and the censoring indicators are missing at random, construct the composite quantile regression estimators of regression function based on the local polynomial method,and establish the asymptotic normality of these estimators, which extends the results of the Kai, et al.(2010) to right-censored data with the censoring indicators missing at random. The simulation studies show that our estimators perform better than the kernel estimation proposed by Wang and Zheng(2014), especially when the error is the heavy tail distribution.
引文
[1] Fan J, Gijbels I. Local Polynomial Modelling and Its Applications. London:Chapman&Hall,1996.
[2] Kai B, Li R, Zou H. Local composite quantile regression smoothing:An efficient and safe alternative to local polynomial regression. J. Roy. Statist. Soc. Ser.B, 2010, 72:49-69.
[3] Fan J, Gijbels I. Censored regression:Local linear approximations and their applications. J.Amer. Statist. Assoc., 1994, 89:560-570.
[4] Guessoum Z, Ould-Said E. On nonparametric estimation of the regression function under random censorship model. Statist. Decisions, 2008, 26:159-177.
[5] Li X, Wang Q. The weighted least square based estimators with censoring indicators missing at random. J. Statist. Plann. Inference, 2012, 142:2913-2925.
[6] Wang J, Zheng M. Nonparametric regression estimation with missing censoring indicators. Chinese Journal of Applied Probability and Statistics, 2014, 30:476-490.
[7] Loprinzi C L, Laurie J A, Wieand H S, et al. Prospective evaluation of prognostic variables from patient-completed questionnaires. North Central Cancer Treatment Group. Journal of Clinical Oncology Official Journal of the American Society of Clinical Oncology, 1994, 12(3):601-607.
[8] Knight K. Limiting distributions for L_1 regression estimators under general conditions. Ann.Statist., 1998, 26:755-770.
[9] Parzen E. On estimation of a probability density function and model. Ann. Math. Statist., 1962,33:1065-1076.
[10] Koenker R. Quantile Regression. Cambridge:Cambridge University Press, 2015.
[2] Kai B, Li R, Zou H. Local composite quantile regression smoothing:An efficient and safe alternative to local polynomial regression. J. Roy. Statist. Soc. Ser.B, 2010, 72:49-69.
[3] Fan J, Gijbels I. Censored regression:Local linear approximations and their applications. J.Amer. Statist. Assoc., 1994, 89:560-570.
[4] Guessoum Z, Ould-Said E. On nonparametric estimation of the regression function under random censorship model. Statist. Decisions, 2008, 26:159-177.
[5] Li X, Wang Q. The weighted least square based estimators with censoring indicators missing at random. J. Statist. Plann. Inference, 2012, 142:2913-2925.
[6] Wang J, Zheng M. Nonparametric regression estimation with missing censoring indicators. Chinese Journal of Applied Probability and Statistics, 2014, 30:476-490.
[7] Loprinzi C L, Laurie J A, Wieand H S, et al. Prospective evaluation of prognostic variables from patient-completed questionnaires. North Central Cancer Treatment Group. Journal of Clinical Oncology Official Journal of the American Society of Clinical Oncology, 1994, 12(3):601-607.
[8] Knight K. Limiting distributions for L_1 regression estimators under general conditions. Ann.Statist., 1998, 26:755-770.
[9] Parzen E. On estimation of a probability density function and model. Ann. Math. Statist., 1962,33:1065-1076.
[10] Koenker R. Quantile Regression. Cambridge:Cambridge University Press, 2015.