Deconvolving a Density from Partially Contaminated Observations

We consider the problem of estimating a continuous bounded probability density function when independent data X1, ..., Xn from the density are partially contaminated by measurement error. In particular, the observations Y1, ..., Yn are such that P(Yj = Xj) = p and P(Yj = Xj + ϵj) = 1 − p, where the errors ϵj are independent (of each other and of the Xj) and identically distributed from a known distribution. When p = 0 it is well known that deconvolution via kernel density estimators suffers from notoriously slow rates of convergence. For normally distributed ϵj the best possible rates are of logarithmic order pointwise and in mean square error. In this paper we demonstrate that for merely partially(0 < p <1) contaminated observations (where of course it is unknown which observations are contaminated and which are not) under mild conditions almost sure rates of order O(((log h−1)/nh)1/2) with h = h(n) = const(log n/n)1/5 are achieved for convergence in L∞-norm. This is equal to the optimal rate available in ordinary density estimation from direct uncontaminated observations (p = 1). A corresponding result is obtained for the mean integrated squared error.