Abstract
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.
Highlights
Density estimation is a widely adopted tool for multiple tasks in statistical inference, machine learning, visualization and exploratory data analysis
Bernstein polynomial and a finite Gaussian mixture model to construct a semi-parametric density estimator, which improves the approximation at the edges
We propose a shrinkage estimator of a density function based on the Bernstein density estimator and using a finite Gaussian mixture density
Summary
Density estimation is a widely adopted tool for multiple tasks in statistical inference, machine learning, visualization and exploratory data analysis. In the non-parametric framework, several methods have been set forward for the smooth estimation of density and distribution functions. It is noteworthy that the Gaussian mixture model can be used to estimate any density function, without any problem of estimation on the edge. The problem at the edge does not arise for the parametric model For this reason, the basic idea of this work is to consider a shrinkage method using Bernstein (Vitale’s estimator) and Gaussian mixture estimators, to construct a shrinkage density estimator, in order to improve the approximation at the edge.
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