Page 106 - Textos de Matemática Vol. 47

96 A. C. ROSA AND M. E. NOGUEIRA
More recently, the nonparametric estimation of conditional quantiles has been extended, in the same framework, to include functional covariates. For a general introduction to the subject, we refer to Ferraty and Vieu [18]. Several authors analysed the asymptotic properties of standard estimators, establi- shing almost complete consistency (Ferraty et al. [17]), normality (Ezzahrioui and Ould-Sa¨ıd [16]) and Lp norm consistency (Laksaci et al. [26], Dabo-Niang and Laksaci [9]).
As far as we know, there are remarkably few results under a general depen- dence condition of ergodicity (cf. Delecroix et al. [11], Delecroix and Rosa [12], Yakowitz et al. [29], La¨ıb and Ould-Sa¨ıd [23]). However, after the work of La¨ıb and Louani [24], there has been a renewed interest in this subject, motivating some new consistency results with rate of convergence (cf. La¨ıb and Louani [25], Chaubey et al. [8], Chaouch and Khardani [7]).
With regard to continuous time stationary processes, Banon [1], inspired by the discrete case, was the first author to propose a kernel density estimator based on an observed sample (Xt,0  t  T) of a di↵usion process X. Apart from di↵usion processes, we point out the pioneering work of Delecroix [10], who obtained rates of convergence for the MSE and the supremum norm cri- teria for a large class of density estimators, as well as the historical paper of Castellana and Leadbetter [6], addressing the convergence rate and the asymptotic normality of an estimator built by the delta method, in the set- ting of mixing processes. Since then, nonparametric estimation for continuous time processes has been extensively studied, proceeding roughly along the same line as for time series. Again, the asymptotic normality and improved rates of convergence of density and regression function estimators were obtained for several convergence criteria. Let us mention, in this context, the surveys of Bosq [4], Bosq and Blanke [5], and the references therein, as well as the works of Lejeune [27], focused on the histogram and the regressogram, and Blanke and Bosq [3], devoted to the regression function. Notice that the referred results were established by imposing mixing conditions on the underlying process, that are not satisfied in many cases, as pointed out by Didi and Louani [14]. We find in Didi and Louani [14, 15] interesting results concerning almost sure con- vergence of the kernel density and regression estimators, both pointwise and uniform, with nonparametric rates, under a continuous version of the ergodic hypothesis considered by Delecroix et al. [11]. In the same setting, Didi [13] proved the strong uniform consistency of a predictor based on the conditional mode.
Following the work of the previous authors, we study, in the present work, the consistency of a kernel type estimator of the conditional distribution function.