Bivariate Uniform Deconvolution

TL;DR

This paper develops a new density estimation method for bivariate uniform deconvolution, deriving four inversion formulas, constructing estimators, and combining them optimally, with theoretical analysis and simulations.

Contribution

It introduces four inversion formulas for bivariate uniform deconvolution and an optimal convex combination of estimators, advancing existing deconvolution techniques.

Findings

Derived four inversion formulas for the density

Constructed four kernel-based estimators

Proposed an asymptotically optimal estimator

Abstract

We construct a density estimator in the bivariate uniform deconvolution model. For this model we derive four inversion formulas to express the bivariate density that we want to estimate in terms of the bivariate density of the observations. By substituting a kernel density estimator of the density of the observations we then get four different estimators. Next we construct an asymptotically optimal convex combination of these four estimators. Expansions for the bias, variance, as well as asymptotic normality, are derived. Some simulated examples are presented.