左截断相依数据下条件分位数的双核局部线性估计
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摘要
本文在左截断相依数据下,利用局部线性估计的方法,先提出了条件分布函数的双核估计;然后利用该估计导出了条件分位数的双核局部线性估计,并建立了这些估计的渐近正态性结果;最后,通过模拟显示该估计在偏移和边界点调节上要比一般的核估计更好.
We construct a double-kernel estimator of conditional distribution function by the local linear approach for left-truncated and dependent data, from which we derive the weighted double-kernel local linear estimator of conditional quantile. The asymptotic normality of the proposed estimators are also established. Finite-sample performance of the estimator is investigated via simulation, and is better than the general kernel estimation in bias and adaptation of edge effects.
We construct a double-kernel estimator of conditional distribution function by the local linear approach for left-truncated and dependent data, from which we derive the weighted double-kernel local linear estimator of conditional quantile. The asymptotic normality of the proposed estimators are also established. Finite-sample performance of the estimator is investigated via simulation, and is better than the general kernel estimation in bias and adaptation of edge effects.
引文
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[2] Cai Z., Wang X., Nonparametric estimation of conditional VaR and expected shortfall, J. Econometrics,2008, 147:120-130.
[3] Cai T., Wei L. J., Wilcox M., Semiparametric regression analysis for clusetered survival data, Biometrics,2000, 87:867-878.
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[7] Ghouch A. E., Keilegom I. V., Local linear quantile regression with dependent censored data, Statist. Sinica,2009, 19:1621-1640.
[8] Hall P., Heyde C. C., Martingale Limit Theory and Its Application, Academic Press, New York, 1980.
[9] He S., Yang G., Estimation of the truncation probability in the random truncation model, Ann Statist.,1998, 26:1011-1027.
[10] Kang S. S., Koehler K. J., Modification of the greenwood formula for correlated response times, Biometrics,1997, 53:885-899.
[11] Lemdani M., Ould-Said E., Poulin N., Asymptotic properties of a conditional quantile estimator with randomly truncated data, J. Multivariate Anal., 2009, 100:546-559.
[12] Liang H. Y., Baek J., Asymptotic normality of conditional density estimation with left-truncated and dependent data, Stat Papers, 2016, 57:1-20.
[13] Liang H. Y., de Una-Alvarez J., Iglesias-Perez M C., Local polynomial estimation of a conditional mean function with dependent truncated data, Test., 2011, 20(3):653-677.
[14] Lynden-Bell D., A method of allowing for known observational selection in small samples applied to 3CR quasars, Monthly Notices of the Royal Astronomical Society, 1971, 155:95-118.
[15] Stute W., Almost sure representations of the product-limit estimator for truncated data, Ann. Statist., 1993,21:146-156.
[16] Volkonskii V. A., Rozanov Y. A., Some limit theorems for random functions, Theory Probab. Appl., 1959,4:178-197.
[17] Wang J. F., Liang H. Y., Fan G. L., Asymptotic properties of conditional quantile estimator under lefttruncated and a-mixing conditions, Commun. Statist. Theory Methods, 2011, 40:2462-2486.
[18] Withers C. S., Conditions for linear processes to be strong mixing, Z Wahrsch verw Gebiete., 1981, 57:477-480.
[19] Xiang X., A kernel estimator of a conditional quantile, J. Multivariate Anal., 1996, 59:206-216.
[20] Yu K., Jones M. C., Local linear quantile regression, J. Amer. Statist. Assoc., 1998, 93:228-237.
[2] Cai Z., Wang X., Nonparametric estimation of conditional VaR and expected shortfall, J. Econometrics,2008, 147:120-130.
[3] Cai T., Wei L. J., Wilcox M., Semiparametric regression analysis for clusetered survival data, Biometrics,2000, 87:867-878.
[4] Dabrowska D., Nonparametric quantile regression with censored data, Sankhya., 1992, 54:252-259.
[5] Doukhan P., Mixing:Properties and Examples, Lecture Notes in Statistics, Springer, Berlin, 1994.
[6] Fan J., Yao Q., Tong H., Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems, Biometrika, 1996, 83:189-206.
[7] Ghouch A. E., Keilegom I. V., Local linear quantile regression with dependent censored data, Statist. Sinica,2009, 19:1621-1640.
[8] Hall P., Heyde C. C., Martingale Limit Theory and Its Application, Academic Press, New York, 1980.
[9] He S., Yang G., Estimation of the truncation probability in the random truncation model, Ann Statist.,1998, 26:1011-1027.
[10] Kang S. S., Koehler K. J., Modification of the greenwood formula for correlated response times, Biometrics,1997, 53:885-899.
[11] Lemdani M., Ould-Said E., Poulin N., Asymptotic properties of a conditional quantile estimator with randomly truncated data, J. Multivariate Anal., 2009, 100:546-559.
[12] Liang H. Y., Baek J., Asymptotic normality of conditional density estimation with left-truncated and dependent data, Stat Papers, 2016, 57:1-20.
[13] Liang H. Y., de Una-Alvarez J., Iglesias-Perez M C., Local polynomial estimation of a conditional mean function with dependent truncated data, Test., 2011, 20(3):653-677.
[14] Lynden-Bell D., A method of allowing for known observational selection in small samples applied to 3CR quasars, Monthly Notices of the Royal Astronomical Society, 1971, 155:95-118.
[15] Stute W., Almost sure representations of the product-limit estimator for truncated data, Ann. Statist., 1993,21:146-156.
[16] Volkonskii V. A., Rozanov Y. A., Some limit theorems for random functions, Theory Probab. Appl., 1959,4:178-197.
[17] Wang J. F., Liang H. Y., Fan G. L., Asymptotic properties of conditional quantile estimator under lefttruncated and a-mixing conditions, Commun. Statist. Theory Methods, 2011, 40:2462-2486.
[18] Withers C. S., Conditions for linear processes to be strong mixing, Z Wahrsch verw Gebiete., 1981, 57:477-480.
[19] Xiang X., A kernel estimator of a conditional quantile, J. Multivariate Anal., 1996, 59:206-216.
[20] Yu K., Jones M. C., Local linear quantile regression, J. Amer. Statist. Assoc., 1998, 93:228-237.