Generalized kernel density estimation with limited data contamination

Abstract

This paper treats the deconvolution model $Y=X+(1-S)U$, where $X$ has Lebesgue density $f_X$, $U$ has a known distribution, and $S$ has a known Bernoulli distribution with $P[S=1]\in (0,1)$. The aim is to estimate $f_X$ using observations from $Y$. Unlike the classic deconvolution model where $P[S=1]=0$ (Fan 1991, Annals of Statistics), this estimation problem is well-posed. This substantially reduces the difficulty of the estimation problem. Existing estimators require the characteristic function of $U$ to be real-valued or else $P[S=1]>1/2$ but the implementation in that case requires selecting three tuning parameters which is not very appealing in applied works. I present an easily implementable nonparametric methodology which removes these restrictions concerning the distribution of $(X,U,S)$. If $U$ is noisy in a certain sense or else if $P[S=1]>1/2$ then a target density belonging to the classic Holder class can be identified as the solution of a well-posed Fredholm integral equation of the second kind. The proposed estimator has a Mean Integrated Square Error converging at a rate which is equal to the optimal nonparametric rate without data contamination (i.e. if $P[S=1]=1$). Moreover, if the distribution of $S$ is unknown, then a feasible estimator is proposed assuming that either the first moment or the second moment of $X$ is known. The feasible estimator has an Integrated Square Error displaying the same speed of convergence in probability. A Monte Carlo experiment reveals good finite-sample properties for the proposed estimators when the distribution of $U$ is supersmooth or skewed