Asymptotic normality for deconvolution kernel density estimators from random fields

TL;DR

This paper establishes the asymptotic normality of deconvolution kernel density estimators for multivariate random fields with spatial interactions, extending previous results from univariate i.i.d. cases to more complex spatial data.

Contribution

It introduces asymptotic normality results for deconvolution kernel density estimators applied to multivariate random fields with spatial dependence, a novel extension from univariate independent cases.

Findings

Proves asymptotic normality under certain conditions.

Extends deconvolution density estimation to spatially dependent data.

Addresses multivariate random fields with known error distributions.

Abstract

The paper discusses the estimation of a continuous density function of the target random field $X_{\bf{i}}$, $\bf{i}\in \mathbb {Z}^N$ which is contaminated by measurement errors. In particular, the observed random field $Y_{\bf{i}}$, $\bf{i}\in \mathbb {Z}^N$ is such that $Y_{\bf{i}}=X_{\bf{i}}+\epsilon_{\bf{i}}$, where the random error $\epsilon_{\bf{i}}$ is from a known distribution and independent of the target random field. Compared to the existing results, the paper is improved in two directions. First, the random vectors in contrast to univariate random variables are investigated. Second, a random field with a certain spatial interactions instead of i. i. d. random variables is studied. Asymptotic normality of the proposed estimator is established under appropriate conditions.