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Abstract: We consider the problem of estimating the unconditional distribution of apost-model-selection estimator. The notion of a post-model-selection estimatorhere refers to the combined procedure resulting from first selecting a modele.g., by a model selection criterion like AIC or by a hypothesis testingprocedure and then estimating the parameters in the selected model e.g., byleast-squares or maximum likelihood, all based on the same data set. We showthat it is impossible to estimate the unconditional distribution withreasonable accuracy even asymptotically. In particular, we show that noestimator for this distribution can be uniformly consistent not even locally.This follows as a corollary to local minimax lower bounds on the performanceof estimators for the distribution; performance is here measured by theprobability that the estimation error exceeds a given threshold. These lowerbounds are shown to approach 1-2 or even 1 in large samples, depending on thesituation considered. Similar impossibility results are also obtained for thedistribution of linear functions e.g., predictors of the post-model-selectionestimator.



Autor: Hannes Leeb, Benedikt M. Poetscher

Fuente: https://arxiv.org/







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