Mean square convergence for estimators of additive regression under random censorship

Testing additivity in nonparametric regression under random censorship

“…We will make frequent use of the following lemma, which was established in Debbarh and Viallon (2007a). We will also make frequent use of the following result, due to Földes and Rejtő (1981).…”

Uniform limit laws of the logarithm for estimators of the additive regression function in the presence of right censored data

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It has been recently shown that nonparametric estimators of the additive regression function could be obtained in the presence of right censoring by coupling the marginal integration method with initial kerneltype Inverse Probability of Censoring Weighted estimators of the multivariate regression function [10]. In this paper, we get the exact rate of strong uniform consistency for such estimators.

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“…estimators. Here, following the ideas introduced in [10], we make use of the marginal integration method, coupled with initial kernel-type I.P.C.W. estimators to provide an estimator for the additive censored regression function.…”

“…This combination leads to estimators for which the theory is easier to derive, which was wanted here, given the technicalities in the proof, even in this simplified setting (note however that, as already mentioned, extensions to other synthetic data estimators can be obtained; see Paragraph 3.1). In a previous work [10], the mean-square convergence rate was established for the integrated estimator defined in (2.7) below. In the present paper, we get the exact corresponding rate of strong uniform consistency (see Theorem 3.2 below).…”

Convergence uniforme d'un estimateur de la fonction de régression additive en données censurées

“…. , d. Debbarh et Viallon [6] ont obtenu la vitesse de convegence en moyenne quadratique de l'estimateur de la régression additive en données censurées défini en (8) à partir de la méthode d'intégration marginale. Dans le même cadre, et sous des conditions plus géné-rales sur la fonction ψ (voir A(ii) ci-après) nous permettant notamment de traiter le cas ψ(y) = y et donc d'appliquer nos résultats à la fonction de régression classique, nous nous proposons d'établir la vitesse de convergence uniforme presque sûre.…”