Local Orthogonal Polynomial Expansion for Density Estimation

TL;DR

LOrPE is a new density estimation method that combines advantages of KDE, LLDE, and OSDE, providing bias reduction at boundaries and improved convergence, especially for densities with sharp truncations.

Contribution

The paper introduces LOrPE, a novel local orthogonal polynomial expansion method that generalizes existing density estimators and improves boundary bias correction.

Findings

LOrPE outperforms KDE, LLDE, and OSDE in simulations for densities with sharp boundaries.

LOrPE achieves boundary bias reduction through kernel reshaping.

Faster asymptotic convergence rates are observed with LOrPE.

Abstract

A Local Orthogonal Polynomial Expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localized expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalizes straightforwardly to multivariate settings. By manner of construction, it is similar to Local Likelihood Density Estimation (LLDE). In the limit of small bandwidths, LOrPE functions as Kernel Density Estimation (KDE) with high-order (effective) kernels inherently free of boundary bias, a natural consequence of kernel reshaping to accommodate endpoints. Faster asymptotic convergence rates follow. In the limit of large bandwidths, LOrPE is equivalent to Orthogonal Series Density Estimation (OSDE) with Legendre polynomials. We compare the performance of LOrPE to KDE, LLDE, and OSDE, in a number of simulation studies. In terms of mean integrated squared error, the results suggest that with a proper balance of the two tuning parameters, bandwidth and degree, LOrPE generally outperforms these competitors when estimating densities with sharply truncated supports.