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The paper focuses on general properties of parametric minimum contrastestimators. The quality of estimation is measured in terms of the rate functionrelated to the contrast, thus allowing to derive exponential risk boundsinvariant with respect to the detailed probabilistic structure of the model.This approach works well for small or moderate samples and covers the case of amisspecified parametric model. Another important feature of the presentedbounds is that they may be used in the case when the parametric set isunbounded and non-compact. These bounds do not rely on the entropy or coveringnumbers and can be easily computed. The most important statistical factresulting from the exponential bonds is a concentration inequality which claimsthat minimum contrast estimators concentrate with a large probability on thelevel set of the rate function. In typical situations, every such set is aroot-n neighborhood of the parameter of interest. We also show that theobtained bounds can help for bounding the estimation risk, constructingconfidence sets for the underlying parameters. Our general results areillustrated for the case of an i.i.d. sample. We also consider several popularexamples including least absolute deviation estimation and the problem ofestimating the location of a change point. What we obtain in these examplesslightly differs from the usual asymptotic results presented in statisticalliterature. This difference is due to the unboundness of the parameter set anda possible model misspecification.

Exponential bounds for minimum contrast estimators