Analysis and conditional optimization of projection estimates for distribution of random variable using Legendre polynomials

TL;DR

This paper introduces algorithms that use Legendre polynomials for improved density and distribution function estimation, optimizing accuracy and computational efficiency through conditional optimization, and compares these methods to traditional histograms.

Contribution

It presents novel algorithms for projection estimates of density and distribution functions using Legendre polynomials with an optimization approach to enhance accuracy and efficiency.

Findings

Algorithms outperform histograms in accuracy for smooth densities.

Conditional optimization improves approximation quality.

Tests show effectiveness across different density smoothness levels.

Abstract

Algorithms for jointly obtaining projection estimates of the density and distribution function of a random variable using Legendre polynomials are proposed. For these algorithms, a problem of the conditional optimization is solved. Such optimization allows one to increase the approximation accuracy with minimum computational costs. The proposed algorithms are tested on examples with different degrees of smoothness of the density. A projection estimate of the density is compared to a histogram that is often used in applications to estimate distributions.