Strong consistency of local linear estimation of a conditional density function under random censorship
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Arabian Journal of Mathematics
Abdelkader Benkhaled · Fethi Madani
· Salah Khardani
Strong consistency of local linear estimation of a conditional density function under random censorship
Received: 4 May 2019 / Accepted: 15 April 2020 © The Author(s) 2020
Abstract In this paper, we study nonparametric local linear estimation of the conditional density of a randomly censored scalar response variable given a functional random covariate. We establish under general conditions the pointwise almost sure convergence with rates of this estimator under α-mixing dependence. Finally, to show interests of our results, on the practical point of view, we have conducted a computational study, first on a simulated data and, then on some real data concerning Kidney transplant data. Mathematics Subject Classification
62G05 · 62G07 · 62G08 · 62G35 · 62G20
1 Introduction Conditional density plays an important role; not only in exploring relationships between responses and covariates, but also in financial econometrics (see Ait-Sahalia [1]). A vast variety of papers use the estimators of conditional densities as building blocks. These papers include those of Robinson [20], Tjøstheim [23], among others. However, in all of these papers, the conditional density function is indirectly estimated. Hyndman et al. [13] have studied the kernel estimator of conditional density estimator and its bias corrected version. There are many advantages of using local linear regression, such as the lack of boundary modifications, high minimax efficiency, and ease of implementation. Then, Bashtannyk and Hyndman [3] have suggested several simple and useful rules for selecting bandwidths for conditional density estimation. Hall et al. [11] applied the crossvalidation technique to estimate the conditional density. Fan and Yim [9] proposed a consistent data-driven bandwidth selection procedure for estimating the conditional density functions. In the last decade, the kernel method has been largely used for nonparametric functional data study; in this context, we refer to the monograph of Ferraty and Vieu [10]. Since then, many interesting publications have appeared. According to the literature, results on the local linear modeling in the functional data setting are limited. Baillo and Grané [4] proposed a local linear estimator (LLE) of the regression operator when the explanatory variable takes values in a Hilbert space, and then when the explanatory variable takes values in a semi-metric space. Demongeot et al. [7] presented local linear estimation of the conditional density when A. Benkhaled University Djilali Liabes of Sidi Bel Abbes, Sidi Bel Abbes, Algeria E-mail: [email protected] F. Madani (B) Laboratory of Stochastic Models, Statistic and Applications (LMSSA), University Tahar Moulay of Saïda, Saïda, Algeria E-mail: [email protected] S. Khardani LMPA, ULCO, University of Lille, Lille, France E-mail: [email protected]
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the data are functional. Furthermore, Messaci et al. [19] used the same approach to estim
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