Abstract
Let (X′i,Y i)′ be a set of observations form a stationary α-mixing process and Θ(x) be the conditional α-th quantile of Y given X = x. Several authors considered nonparametric estimation of Θ(x) in the i.i.d. setting. Assuming the smoothness of ΘFF(x), we estimate it by local polynomial fitting and prove the asymptotic normality and the uniform convergence.
Similar content being viewed by others
References
Babu, G. J. (1989). Strong Representation for LAD estimators in linear models, Probab. Theory. Related Fields, 83, 547–558.
Bhattacharya, P. K. and Gangopadhyay, A. (1990). Kernel and nearest neighbor estimation of a conditional quantile, Ann. Statist., 18, 1400–1415.
Chaudhuri, P. (1991a). Nonparametirc estimates of regression quantiles and their local Bahadur representation, Ann. Statist., 19, 760–777.
Chaudhuri, P. (1991b). Global nonparametirc estimation of conditional quantile functions and their derivatives, J. Multivariate Anal., 39, 246–269.
Chaudhuri, P., Doksum, K. and Samarov, A. (1997). On average derivative quantile regression, Ann. Statist., 25, 715–744.
Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications, Chapman & Hall, London.
Fan, J., Hu, T.-C. and Truong, Y. K. (1994). Robust nonparametric function stimation, Scand. J. Statist., 21, 433–446.
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications, Academic Press, San Diego.
Jones, M. C. and Hall, P. (1990). Mean squared error properties of kernel estimates of regression quantiles, Statist. Probab. Lett., 10, 283–289.
Liebscher, E. (1996). Strong convergence of surns of α-mixing random variables with applications to density estimation, Stochastic Process. Appl., 65, 69–80.
Masry, E. (1996a). Multivariate local polynomial regression for time series: uniform strong consistency and rates, J. Time Ser. Anal., 17, 571–599.
Masry, E. (1996b). Multivariate regression estimation: local polynomial fitting for time series, Stochastic Processes Appl., 65, 81–101.
Masry, E. and Fan, J. (1997). Local plynomial estimation of regression functions for mixing processes, Scand. J. Statist., 24, 166–179.
Mehra, K. L., Rao, M. S. and Upadrasta, S. P. (1991). A smooth conditional quantile estimator and related applications of conditional empirical processes, J. Multivariate Anal., 37, 151–179.
Rio, E. (1995). The functional law of the iterated logarithm for strongly mixing sequences, Ann. Probab., 23, 1188–1203.
Truong, Y. K. and Stone, C. J. (1992). Nonparametric function estimation involving time series, Ann. Statist., 20, 77–97.
Welsh A. H. (1996). Robust estimation of smooth regression and spread functions and their derivatives, Statist. Sinica, 6, 347–366.
Xiang, X. (1996). A kernel estimator of a conditional quantile, J. Multivariate Anal., 59, 206–216.
Author information
Authors and Affiliations
About this article
Cite this article
Honda, T. Nonparametric Estimation of a Conditional Quantile for α-Mixing Processes. Annals of the Institute of Statistical Mathematics 52, 459–470 (2000). https://doi.org/10.1023/A:1004113201457
Issue Date:
DOI: https://doi.org/10.1023/A:1004113201457