Deconvolving Multivariate Density from Random Field

We consider the problem of estimating a continuous bounded multivariate probability density function (pdf) when the random field Xi, i ∈ Zd from the density is contaminated by measurement errors. In particular, the observations Yi, i ∈ Zd are such that Yi = Xi + εi, where the errors εi are a sample from a known distribution. We improve the existing results in at least two directions. First, we consider random vectors in contrast to most existing results which are only concerned with univariate random variables. Secondly, and most importantly, while all the existing results focus on the temporal cases (d = 1), we develop the results for random vectors with a certain spatial interaction. Precise asymptotic expressions and bounds on the mean-squared error are established, along with rates of both weak and strong consistencies, for random fields satisfying a variety of mixing conditions. The dependence of the convergence rates on the density of the noise field is also studied.