Likelihood Ratio Tests in Contamination Models

We study the asymptotic distribution of the likelihood ratio statistic to test whether the contamination of a known density f₀ by another density of the same parametric family reduces to f₀. The classical asymptotic theory for the likelihood ratio statistic fails, and we propose a general reparametrization which ensures regularity properties. Under the null hypothesis, the likelihood ratio statistic converges to the supremum of a squared truncated Gaussian process. The result is extended to the case of the contamination of a mixture of p known densities by q other densities of the same family.