Moderate deviations of minimum contrast estimators under contamination.

Since statistical models are simplifications of reality, it is important in estimation theory to study the behavior of estimators also under distributions (slightly) different from the proposed model. In testing theory, when dealing with test statistics where nuisance parameters are estimated, knowledge of the behavior of the estimators of the nuisance parameters is needed under alternatives to evaluate the power. In this paper the moderate deviation behavior of minimum contrast estimators is investigated not only under the supposed model, but also under distributions close to the model. A particular example is the (multivariate) maximum likelihood estimator determined within the proposed model. The set-up is quite general, including also, for instance, discrete distributions. The rate of convergence under alternatives is determined both when comparing the minimum contrast estimator with a "natural" parameter in the parameter space and when comparing it with the proposed "true" value in the parameter space. It turns out that under the model the asymptotic optimality of the maximum likelihood estimator in the local sense continues to hold in the moderate deviation area.