Elias Ould-Saïd

Asymptotic normality of a nonparametric estimator of the conditional mode function for functional data

Abstract We consider the estimation of the conditional mode function when the covariables take values in some abstract function space. It is shown that, under some regularity conditions, the kernel estimate of the conditional mode is asymptotically normally distributed. From this, we derive the asymptotic normality of a predictor and propose confidence bands for the conditional mode function. Simulations are drawn to show how our methodology can be implemented.

Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis

Let (X n ,Y n ) n ≤1 be a R d ×R valued stationary process. Define the estimator of the conditional mode of Y 1 given X 1=x as the random variable θ n (x) that maximizes a kernel estimator of the conditional density of Y 1 given X 1 = x. We establish asymptotic normality of θ n (x) when the process (X n ,Y n ) n ≤1 is assumed to be strongly mixing. We derive from our results asymptotic normality of a predictor and propose a confidence bands for the conditional mode function. A simulation study shows how good the normality of the conditional mode function estimator is when dealing with samples of finite sizes.

Asymptotic Results of a Nonparametric Conditional Quantile Estimator for Functional Time Series

We consider the estimation of the conditional quantile function when the covariates take values in some abstract function space. The main goal of this article is to establish the almost complete convergence and the asymptotic normality of the kernel estimator of the conditional quantile under the α-mixing assumption and on the concentration properties on small balls of the probability measure of the functional regressors. Some applications and particular cases are studied. This approach can be applied in time series analysis to the prediction and building of confidence bands. We illustrate our methodology with El Niño data.

Strong uniform consistency of nonparametric estimation of the censored conditional mode function

Let (T n ) n≥1 be a sequence of independent and identically distributed of interest random variables and (X n ) n≥1 be a sequence of covariates. In the censorship model, the random variable T is subject to random censoring by another random variable C. Let Θ(x) be the conditional mode function of T given X = x. In this article, we define a new kernel estimator Θ n (x) of Θ(x) and we establish the uniform strong consistency with a rate of convergence.

Asymptotic properties of a conditional quantile estimator with randomly truncated data

Let Y be a response variable that is subject to left-truncation by a variable T. We consider the problem of estimating its conditional quantile function given a vector of covariates X. We derive almost sure (a.s.) consistency and asymptotic normality results for a kernel estimate of the conditional quantile function. Simulations are drawn to illustrate the results for finite samples.

Asymptotic Distribution of Robust Estimator for Functional Nonparametric Models

We propose a family of robust nonparametric estimators for regression function based on kernel method. We establish the asymptotic normality of the estimator under the concentration properties on small balls of the probability measure of the functional explanatory variables. Useful applications to prediction, discrimination in a semi-metric space, and confidence curves are given. In addition, to highlight the generality of our purpose and to emphasize the role of each of our hypotheses, several special cases of our general conditions are also discussed. Finally, some numerical study in chemiometrical real data are carried out to compare the sensitivity to outliers between the classical and robust regression.

Convergence of the conditional Kaplan-Meier estimate under strong mixing

We establish the uniform almost complete consistency of the Kaplan-Meier conditional estimate, for stationary strong mixing processes, in the presence of right censoring.

Strong Consistency Rate for the Kernel Mode Estimator Under Strong Mixing Hypothesis and Left Truncation

Let (Y N ) N≥1 be a sequence of copies of a random variable of interest Y. In this article, we study the kernel estimator, say , of the mode of Y when Y is subject to the random left truncation. While based on n (n ≤ N) actual observations fulfilling the well-known α-mixing condition, we establish the strong consistency with a rate of the proposed estimator .

Asymptotic behavior of the hazard rate kernel estimator under truncated and censored data

In this article, we give the asymptotic mean integrated squared error and the mean squared error for the kernel estimator of the hazard rate from truncated and censored data. Martingale techniques and combinatory calculus are used to obtain these results. A probability bound and the optimal bandwidth choice are also given.

Strong Approximation of Quantile Function for Strong Mixing and Censored Processes

ABSTRACT Let (X i ) i≥1 be a sequence of strong-mixing random variables with common unknown absolutely continuous distribution function F subject to random right censoring. Let F −1(p) denote the pth (p ∈ ]0, 1[) quantile function of the marginal distribution function F of the X i ′s which is estimated by a sample quantile (p). In this article, we derive the strong consistency and a Bahadur-type representation for (p), the quantile function of the Kaplan–Meier estimator of F for strong-mixing processes. Then we extend the result of Cheng (1984) to the dependent case.