Density testing in a contaminated sample

We study non-parametric tests for checking parametric hypotheses about a multivariate density f of independent identically distributed random vectors Z 1 , Z 2 , … which are observed under additional noise with density ψ . The tests we propose are an extension of the test due to Bickel and Rosenblatt [On some global measures of the deviations of density function estimates, Ann. Statist. 1 (1973) 1071–1095] and are based on a comparison of a nonparametric deconvolution estimator and the smoothed version of a parametric fit of the density f of the variables of interest Z i . In an example the loss of efficiency is highlighted when the test is based on the convolved (but observable) density g = f * ψ instead on the initial density of interest f.