Page 105 - Textos de Matemática Vol. 47

A NOTE ON KERNEL ESTIMATION OF THE CONDITIONAL QUANTILE FUNCTION IN CONTINUOUS TIME ERGODIC PROCESSES
ANA CRISTINA ROSA AND MARIA EM´ILIA NOGUEIRA
Dedicated to Nazar´e Mendes Lopes for everything she taught us and for her friendship
Abstract. We prove the strong consistency of a kernel type estimator of the conditional distribution function when the data generation process is time continuous and satisfies a general ergodic hypothesis. The strong consistency of the quantile function corresponding to this estimator is then established.
1. Introduction
It is well known that nonparametric methods provide essential tools for the statistical analysis and applications of stochastic processes, namely the pre- diction of future observations. Although most popular predictors are based on the regression function, other conditional parameters such as the mode and the quantiles have received considerable attention in literature. In particular, the robustness of quantiles to outliers and heavy-tailed error distributions jus- tifies their use as alternatives to the regression function for quantifying the dependence structure between a response variable and a covariate.
Quantile estimates are usually derived either via quantile regression models (cf. Koenker [22], for an overview, Gannoun et al. [20] and Ghouch and Keilegom [21], among others) or by inverting nonparametric estimators of the conditional distribution function. Following this approach, the main results con- cern essentially the strong consistency and asymptotic normality of kernel type estimators and were obtained in the framework of time series under mixing as- sumptions on the data generation process (cf., for instance, Gannoun et al. [19] for complete data and Liang and Un˜a-A´lvarez [28] for censored observations).
Accepted: 17 February 2015.
2010 Mathematics Subject Classification. 62G05, 62M09, 62M20.
Key words and phrases. nonparametric estimation, continuous time processes, ergodic data, kernel method, conditional quantile, strong consistency.
The work was supported by the Department of Mathematics of the University of Coimbra.
95