Memorandum 1598
T. Inglot & W.C.M. Kallenberg
Abstract:
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 for instance also 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.
Keywords:
Minimum contrast estimator, multivariate maximum likelihood estimator, moderate deviation, asymptotic
optimality, alternative, misspecification, nuisance parameter,
robustness, score function
Mathematics Subject Classification: 62F12, 62H12, 62E20, 60F10