Random positive linear operators and their applications to nonparametric statistics

TL;DR

This paper introduces a method using random positive linear operators, specifically random Bernstein polynomials, to construct confidence bands for distribution functions and their derivatives, improving estimation simplicity and performance.

Contribution

It presents a novel application of random positive linear operators in nonparametric estimation, providing explicit confidence bands and intervals with performance advantages.

Findings

Explicit confidence bands for distribution functions using random Bernstein polynomials.

Improved estimation of distribution functions over classical empirical methods.

Performance gains demonstrated by the second-order random polynomial estimator.

Abstract

We outline a general procedure on how to apply random positive linear operators in nonparametric estimation. As a consequence, we give explicit confidence bands and intervals for a distribution function $F$ concentrated on $[0,1]$ by means of random Bernstein polynomials, and for the derivatives of $F$ by using random Bernstein-Kantorovich type operators. In each case, the lengths of such bands and intervals depend upon the degree of smoothness of $F$ or its corresponding derivatives, measured in terms of appropriate moduli of smoothness. In particular, we estimate the uniform distribution function by means of a random polynomial of second order. This estimator is much simpler and performs better than the classical uniform empirical process used in the celebrated Dvoretzky-Kiefer-Wolfowitz inequality.