Deconvolving multidimensional density from partially contaminated observations

Abstract

We consider the problem of estimating a continuous bounded d-variate joint probability density function of ( X 1, X 2,…, X d ) when a stationary process X 1,…, X n from the density are partially contaminated by measurement error. In particular, the observations Y 1,…, Y n are such that P (Y j=X j)=p and P (Y j=X j+ε j)=1−p , where the errors { ε j } are independent (of each other and of { X j }) and identically distributed from a known distribution. We are interested in estimating the joint density of ( X 1, X 2,…, X d ). When p=0, it is well known that deconvolution via kernel density estimators suffers from notoriously slow rate of convergence (Ann. Statist. 18(2) (1990) 806; J. Multivariate Anal. 44 (1993a) 57). In contrast, for univariate partially contaminated observations (0< p<1), almost sure rates of O(( ln n/nh n) 1/2) has been achieved for convergence in L ∞-norm (J. Multivariate Anal. 55 (1995) 246). But in multivariate case, the situation is much more complex because of the dependence among the data. One purpose of this paper is to fill in this void. Furthermore, under mild conditions, we also obtain the asymptotic mean squared error and asymptotic normality of the estimator with partially contaminated observations.