Stochastic approximation algorithms for multivariate functionals estimation with medical and cognitive fields applications.

Résumé

Multivariate recursive estimation is the central focus of this thesis. Our basic objective is to construct multivariate functional estimators using stochastic approximations methods. In the opening section, we provide a general introduction of non-parametric estimation topic and the original recursive stochastic approximation algorithm. In the first chapter, we introduce a multivariate recursive estimator for the distribution function. We study the asymptotic properties of this generalized estimator and we compare it with non-recursive Nadaraya's multivariate distribution estimator. It turns out that, with an adequate choice of the stepsize and an appropriate choice of the bandwidth, the MSE (Mean Squared Error) of the multivariate estimator with plug-in bandwidth selection method can be smaller than the two other estimators, namely the multivariate recursive one with cross-validation selection and the non-recursive one of Nadaraya's estimator. The second chapter deals with non-parametric estimation of a conditional cumulative distribution function (CCDF) $\pi: (y|x)\longmapsto\mathbb{P}\left[Y\leqslant y|X=x\right].$ Using the same recursive approach, we suggest a multivariate recursive estimator defined by stochastic approximation algorithm. We investigate the statistical inferences of our estimator and compare them with those of non-recursive Nadaraya-Watson's estimator. Given the idea of conditional estimation, and for the third chapter, we construct a generalized semi-recursive kernel-type estimator of the regression function $ r_{\varphi}: x \longmapsto \mathbb{E}[\varphi(Y) | X = x],$ for a chosen measurable function $\varphi$ and $x\in \rr^d$. In order to examine the asymptotic properties of this estimator, we first calculate the bias and the variance of our proposed estimator which strongly depend on the choice of three parameters which are the stepsizes and the bandwidth. Moreover, we are interested in studying the strong pointwise convergence rate of our estimator. It turns out that under the estimation by confidence intervals, the proposed estimator is better than that of the non-recursive Nadaraya Watson regression one. As far as the fourth chapter is concerned, we attempt to explore cognitive processes and mental representations enacted when a human being prepares to write a word according to the idea developed in~\cite{perreto19}. For this objective, we set forward a non-parametric multivariate recursive kernel regression estimation under missing data to describe writing word production. We examine the asymptotic properties of our recursive estimator and its performance against the non-recursive one. As application, the proposed estimator is then applied to the behavioral data to classify some participants in groups. This classification can be a departure point to understand written behavior variations. The closing section is devoted for the conclusion part as well as some perspectives for future researches.

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