The central focus of this paper is upon the alleviation of the boundary problem when the probability density function has a bounded support. Mixtures of beta densities have led to different methods of density estimation for data assumed to have compact support. Among these methods, we mention Bernstein polynomials which leads to an improvement of edge properties for the density function estimator. In this paper, we set forward a shrinkage method using the Bernstein polynomial and a finite Gaussian mixture model to construct a semi-parametric density estimator, which improves the approximation at the edges. Some asymptotic properties of the proposed approach are investigated, such as its probability convergence and its asymptotic normality. In order to evaluate the performance of the proposed estimator, a simulation study and some real data sets were carried out.
In this paper, we propose a recursive estimators of the regression function based on the two-time-scale stochastic approximation algorithms and the Bernstein polynomials. We study the asymptotic properties of this estimators. We compare the proposed estimators with the classic regression estimator using the Bernstein polynomial defined by Tenbusch. Results showed that, our proposed recursive estimators can overcome the problem of the edges associated with kernel regression estimation with a compact support. The proposed recursive two-time-scale estimators are compared to the non-recursive estimator introduced by Tenbusch and the performance of the two estimators are illustrated via simulations as well as two real datasets.
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