On A strongly consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function is studied when the sample observations are contaminated with random noise. Previous authors have proposed estimators which use kernel density and deconvolution techniques. The appearance and properties of the previously proposed estimators are affected by constants M_n and h_n which the user may choose. However, the optimal choices of these constants depend on the sample size n, the noise distribution and the unknown distribution which is being estimated. Hence, in practice, M_n and h_n are optimally selected as functions of the data. In this paper it is shown that a class of the proposed estimators are uniformly, strongly consistent when M_n and h_n are allowed to be random variables. Even when M_n and h_n are constants, these results are new findings.