Electronic Journal of Statistics

A strong uniform convergence rate of a kernel conditional quantile estimator under random left-truncation and dependent data

Elias Ould-Saïd, Djabrane Yahia, and Abdelhakim Necir

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In this paper we study some asymptotic properties of the kernel conditional quantile estimator with randomly left-truncated data which exhibit some kind of dependence. We extend the result obtained by Lemdani, Ould-Saïd and Poulin [16] in the iid case. The uniform strong convergence rate of the estimator under strong mixing hypothesis is obtained.

Article information

Electron. J. Statist. Volume 3 (2009), 426-445.

First available in Project Euclid: 26 May 2009

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

Kernel estimator quantile function rate of convergence strong mixing strong uniform consistency truncated data


Ould-Saïd, Elias; Yahia, Djabrane; Necir, Abdelhakim. A strong uniform convergence rate of a kernel conditional quantile estimator under random left-truncation and dependent data. Electron. J. Statist. 3 (2009), 426--445. doi:10.1214/08-EJS306. https://projecteuclid.org/euclid.ejs/1243343760

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  • [1] Boente, G, Fraiman, R (1995) Asymptotic distribution of smoothers based on local means and local medians under dependence. J of Multivariate Anal, 54:77–90.
  • [2] Bosq, D (1998) Nonparametric Statistics for Stochastic Processes: Estimation and Prediction. (Second Edition). Lecture Notes in Statistics, 110, Springer Verlag, New, York.
  • [3] Cai, Z (1998) Kernel density and hazard rate estimation for censored dependent data. J of Multivariate Anal, 67:23–34.
  • [4] Cai, Z (2001) Estimating a distribution function for censored time series data. J Multivariate Anal, 78:299–318.
  • [5] Chaudhuri, P, Doksum, K, Samarov, A (1997) On average derivative quantile regression. Ann Statist, 25:715–744.
  • [6] Doukhan, P (1994) Mixing: Properties and examples. Lecture Notes in Statistics 85, Springer-Verlag, New, York.
  • [7] Feigelson, ED, Babu, GJ (1992) Statistical Challenges in Modern Astronomy. Berlin Heidelberg, Springer Verlag, New, York.
  • [8] Ferraty, F, Vieu, P (2006) Nonparametric Functional Data Analysis, Theory and Practice. Springer-Verlag, New, York.
  • [9] Gannoun, A (1989) Estimation de la médiane conditionnelle. Thèse de doctorat de l’Université de Paris, VI.
  • [10] Gannoun, A (1990) Estimation non paramétrique de la médiane conditionnelle médianogramme et méthode du noyau. Publication de l’Institut de Statistique de l’université de Paris, 11–22.
  • [11] Gürler, U, Stute, W, Wang, JL (1993) Weak and strong quantile representations for randomly truncated data with applications. Statist Probab Lett, 17:139–148.
  • [12] He, S, Yang, G (1994) Estimating a lifetime distribution under different sampling plan. In S.S. Gupta J.O. Berger (Eds.) Statistical decision theory and related topics 5:73–85 Berlin Heidelberg, Springer Verlag, New, York.
  • [13] He, S, Yang, G (1998) Estimation of the truncation probability in the random truncation model. Ann Statist, 26:1011–1027.
  • [14] Lemdani, M, Ould-Saïd, E (2007) Asymptotic behaviorof the hazard rate kernel estimator under truncated and censored data. Comm. in Statist. Theory & Methods, 37-155-174.
  • [15] Lemdani, M, Ould-Saïd, E, Poulin, N (2005) Strong representation of the quantile function for left-truncated and dependent data. Math Meth Statist, 14:332–345
  • [16] Lemdani, M, Ould-Saïd, E, Poulin, N (2008) Prediction for a left truncated model via estimation of the conditional quantile. (In press in J. Multivariate, Anal.).
  • [17] Lynden-Bell, D (1971) A method of allowing for known observational selection in small samples applied to 3CR quasars., Monthly Notices Roy Astron Soc 155:95–118.
  • [18] Masry, E (1986) Recursive probability density estimation for weakly dependent processes. IEEE Transactions on Information Theory, 32:254–267.
  • [19] Masry, E, Tjϕstheim, D (1995) Nonparametric estimation and identification of nonlinear time series. Econometric Theory 11:258–289.
  • [20] Mehra, KL, Rao, MS, Upadrasta, SP (1991) A smooth conditional quantile estimator and related applications of conditional empirical processes. J Multivariate Anal, 37:151–179.
  • [21] Ould-Saïd, E, Lemdani, M (2006) Asymptotic properties of a nonparametric regression function estimator with randomly truncated data. Ann Instit Statist Math, 58:357–378.
  • [22] Ould-Saïd, E, Tatachak, A (2007) Strong uniform consistency rate for the kernel mode under strong mixing hypothesis and left-truncation. (In press in Comm. Statist. Theory &, Method).
  • [23] Rio, E (2000) Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants (In French). Mathématiques et Applications, 31, Springer Verlag, New York.,
  • [24] Roussas, GG (1969). Nonparametric estimation of the transition distribution function of a Markov process. Ann Math Statist, 40:1386–140.
  • [25] Samanta, M (1989) Nonparametric estimation of conditional quantiles., Statist Probab Lett 7, 407–412.
  • [26] Schlee, W (1982) Estimation non paramétrique du, αquantile conditionnel. Statistique et Analyse des données 1:32–47.
  • [27] Stone, C (1977) Consistent nonparametric regression. Ann Statist, 5:595–645.
  • [28] Stute, W (1993) Almost sure representation of the product-limit estimator for truncated data. Ann. Statits., 21:146–156.
  • [29] Tsai, WY, Jewell, NP, Wang, MC (1987). A note on the product-limit estimator under right censoring and left truncation. Biometrika, 74:883–886.
  • [30] Wang, MC, Jewell, NP, Tsai, WY (1986) Asymptotic properties of the product-limit estimate under random truncation. Ann Statist, 14:1597–1605.
  • [31] Woodroofe, M (1985) Estimating a distribution function with truncated data. Ann Statist, 13:163–177.
  • [32] Xiang, X (1996) A kernel estimator of a conditional quantile. J Multivariate Anal, 59:206–216.