Recursive density estimators based on Robbins–Monro’s scheme and using Bernstein polynomials

Abstract

In this paper, we consider the alleviation of the boundary problem when the probability density function has bounded support. We apply Robbins-Monro's algorithm and Bernstein polynomials to construct a recursive density estimator. We study the asymptotic properties of the proposed recursive estimator. We then compared our proposed recursive estimator with many others estimators. Finally, we confirm our theoretical result through a simulation study and then using two real datasets.

Optimal bandwidth selection for recursive Gumbel kernel density estimators

“…We plan also to consider Bernstein polynomials rather than kernels and to propose an adaptation of the estimators developed in [15] and [36] with application to heavy tailed data. Moreover, we plan to make an extensions of our proposed plug-in method in future with application on extreme value and to consider the case of the averaged Révész's regression estimators (see [22] and [31,32]) and the semi-recursive kernel regression estimators proposed by [34] and the case of time series as in [13] in recursive way (see [14]).…”

Adaptive recursive kernel conditional density estimators under censoring data

“…We plan to make an extensions of our proposed estimators by considering Bernstein polynomials rather than kernels and to propose an adaptation of the estimators developed in Jmaei et al (2017) and Slaoui and Jmaei (2019) in the context of conditional density estimation under censoring data.…”

Wild bootstrap bandwidth selection of recursive nonparametric relative regression for independent functional data

“…We plan to extend this work by considering Bernstein polynomials rather than kernels and to propose an adaptation of the estimators developed in Jmaei et al [30] and Slaoui and Jmaei [48] in the case of functional data. We plan also to compare these estimators to the kernel nearest-neighbor approach developed in Kara et al [32], the semi-parametric functional projection pursuit regression [11], the single index model [25], the partial linear models [4,35] and the sparse modeling approach [5].…”