Moderate Deviations of Maximum Likelihood Estimators under Alternatives

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 the (multivariate) maximum likelihood estimator determined within a proposed model is investigated not only under this model, but also under distributions close to the model. The set-up is quite general, including for instance also discrete distributions. It turns out that under the model the asymptotic optimality of the maximum likelihood estimator in the local sense continuous to hold in the moderate deviation area. The rate of convergence under alternatives is determined both when comparing the maximum likelihood estimator with a ”natural” parameter in the parameter space and when comparing it with the proposed ”true” value in the parameter space. The methods can also be applied to general M-estimators.