A penalized exponential risk bound in parametric estimation

The paper offers a novel unified approach to studying the accuracy of parameter estimation by the quasi likelihood method. Important features of the approach are: (1) The underlying model {is not assumed to be parametric}. (2) No conditions on parameter identifiability are required. The parameter set can be unbounded. (3) The model assumptions are quite general and there is no specific structural assumptions like independence or weak dependence of observations. The imposed conditions on the model are very mild and can be easily checked in specific applications. (4) The established risk bounds are {nonasymptotic} and valid for large, moderate and small samples. (5) The main result is the concentration property of the quasi MLE giving an nonasymptotic exponential bound for the probability that the considered estimate deviates out of a small neighborhood of the "true" point. In standard situations under mild regularity conditions, the usual consistency and rate results can be easily obtained as corollaries from the established risk bounds. % The approach and the results are illustrated on the example of generalized linear and single-index models.