Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance

TL;DR

This paper establishes the asymptotic normality of the deconvolution kernel density estimator when the error variance diminishes with sample size, providing theoretical insights and simulation evidence for different asymptotic regimes.

Contribution

It derives the asymptotic normality of the deconvolution kernel density estimator under vanishing error variance, extending existing results to new asymptotic scenarios.

Findings

Asymptotic normality holds under two different conditions relating $\sigma_n$ and $h_n$.

Simulations illustrate the theoretical results and show situations favoring models with decreasing $\sigma_n$.

Models with $\sigma_n o 0$ can outperform fixed $\sigma$ models in certain cases.

Abstract

Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+\sigma_n Z_i$ and the $Y$'s and $Z$'s are independent. Assume that the $Y$'s are unobservable and that they have the density $f$ and also that the $Z$'s have a known density $k.$ Furthermore, let $\sigma_n$ depend on $n$ and let $\sigma_n\to 0$ as $n\to\infty.$ We consider the deconvolution problem, i.e. the problem of estimation of the density $f$ based on the sample $X_1,...,X_n.$ A popular estimator of $f$ in this setting is the deconvolution kernel density estimator. We derive its asymptotic normality under two different assumptions on the relation between the sequence $\sigma_n$ and the sequence of bandwidths $h_n.$ We also consider several simulation examples which illustrate different types of asymptotics corresponding to the derived theoretical results and which show that there exist situations where models with $\sigma_n\to 0$ have to be preferred to the models with fixed $\sigma.$