Estimation of the quantile function using Bernstein–Durrmeyer polynomials

Abstract

This paper studies quantile estimation using Bernstein–Durrmeyer polynomials in terms of its mean squared error and integrated mean squared error including rates of convergence as well as its asymptotic distribution. Whereas the rates of convergence are achieved for i.i.d. samples, we also show that the consistency more or less directly follows from the consistency of the sample quantiles, such that our proposal can also be applied for risk measurement in finance and insurance. Furthermore, an improved estimator based on an error-correction approach is proposed for which a general consistency result is established. A crucial issue is how to select the degree of Bernstein–Durrmeyer polynomials. We propose a novel data-adaptive approach that controls the number of modes of the corresponding density estimator. Its consistency including an uniform error bound as well as its limiting distribution in the sense of a general invariance principle are established. The finite sample properties are investigated by a Monte Carlo study. Finally, the results are illustrated by an application to photovoltaic energy research.