Semiparametric Unimodal Density Estimation Using Bernstein Polynomials

• Activity Number: 336
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Computing
• Author(s): Bradley Turnbull*+ and Sujit Kumar Ghosh
• Companies: North Carolina State University and North Carolina State University
• Keywords: Density Estimation ; Beta Mixture ; Bernstein Polynomials ; R Package ; Quadratic Programming

Abstract

The estimation of probability density functions is one of the fundamental aspects of any statistical inference and most commonly based on an assumed family of parametric models, which are known to be unimodal. Parametric assumptions, however, may not be adequate for many inferential problems. This paper presents a flexible unimodal class of mixture of Beta densities as a sequence of Bernstein Polynomials. We show that the estimation of the mixing weights, and the degree of the polynomial, can be accomplished using a weighted least squares criteria subject to a set of linear constraints. We efficiently compute the number of mixing components and associated mixing weights of the beta mixture using quadratic programming techniques. Simulation studies are conducted to demonstrate the performance of the density estimates in terms of popular functional norms (e.g., Lp norms). The true underlying densities are allowed to be unimodal symmetric and skewed, with finite, infinite or semi-finite supports. An R package was also created to allow the user to input a data set and return the estimated density, distribution, quantile, and random sample generating functions.