Local Estimation of a Multivariate Density and its Derivatives

TL;DR

This paper compares four methods for estimating multivariate densities and their derivatives, analyzing their asymptotic properties and convergence rates, with a focus on local moment matching as the most effective approach.

Contribution

It provides a comprehensive comparison of four local estimation methods for multivariate densities and derivatives, highlighting the superior convergence rate of local moment matching.

Findings

Refined local moment matching achieves the best convergence rates.

Asymptotic properties are derived for all four estimators.

Explicit examples using Gaussian kernels are provided.

Abstract

We analyze four different approaches to estimate a multivariate probability density (or the log-density) and its first and second order derivatives. Two methods, local log-likelihood and local Hyv\"arinen score estimation, are in terms of weighted scoring rules with local quadratic models. The other two approaches are matching of local moments and kernel density estimation. All estimators depend on a general kernel, and we use the Gaussian kernel to provide explicit examples. Asymptotic properties of the estimators are derived and compared. In terms of rates of convergence, a refined local moment matching estimator is the best.