Semiparametric density testing in the contamination model.

Summary In this paper we investigate a semiparametric testing approach to answer if the Gaussian assumption made by McLachlan et al. (2006) on the unknown component of their false discovery type mixture model was a posteriori correct or not. Based on a semiparametric estimation of the Eu-clidean parameters of the model (free from the Gaussian assumption), our method compares pairwise the Hermite coefficients of the model estimated directly from the data with the ones obtained by plugging the estimated parameters into the Gaussian version of the false discovery mixture model. These comparisons are incorporated into a sum of square type statistic which order is controlled by a penalization rule. We prove under mild conditions that our test statistic is asymptotically χ 2 (1)-distributed and study its behavior under different types of alternatives, including contiguous non-parametric alternatives. Several level and power studies are numerically conducted on models close to those considered in McLachlan et al. (2006) to validate the suitability of our approach. We also discuss the lack of power of the maximum likelihood version of our test in a neighborhood of certain non identifiable situations and implement our testing procedure on the three microarray real datasets analyzed in McLachlan et al.