Improved Density and Distribution Function Estimation

TL;DR

This paper introduces improved kernel density and distribution estimators that leverage moment restrictions and generalized empirical likelihood to reduce variance, with applications in semi-parametric models and diagnostic testing.

Contribution

It develops new estimators that incorporate moment restrictions via generalized empirical likelihood, enhancing density and distribution function estimation in semi-parametric models.

Findings

Estimators are consistent under certain conditions.

Asymptotic mean squared error properties are derived.

Simulation shows improved small sample performance.

Abstract

Given additional distributional information in the form of moment restrictions, kernel density and distribution function estimators with implied generalised empirical likelihood probabilities as weights achieve a reduction in variance due to the systematic use of this extra information. The particular interest here is the estimation of densities or distributions of (generalised) residuals in semi-parametric models defined by a finite number of moment restrictions. Such estimates are of great practical interest, being potentially of use for diagnostic purposes, including tests of parametric assumptions on an error distribution, goodness-of-fit tests or tests of overidentifying moment restrictions. The paper gives conditions for the consistency and describes the asymptotic mean squared error properties of the kernel density and distribution estimators proposed in the paper. A simulation study evaluates the small sample performance of these estimators. Supplements provide analytic examples to illustrate situations where kernel weighting provides a reduction in variance together with proofs of the results in the paper.