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October 2006 Can one estimate the conditional distribution of post-model-selection estimators?
Hannes Leeb, Benedikt M. Pötscher
Ann. Statist. 34(5): 2554-2591 (October 2006). DOI: 10.1214/009053606000000821


We consider the problem of estimating the conditional distribution of a post-model-selection estimator where the conditioning is on the selected model. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion such as AIC or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate this distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (local) minimax lower bounds on the performance of estimators for this distribution. Similar impossibility results are also obtained for the conditional distribution of linear functions (e.g., predictors) of the post-model-selection estimator.


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Hannes Leeb. Benedikt M. Pötscher. "Can one estimate the conditional distribution of post-model-selection estimators?." Ann. Statist. 34 (5) 2554 - 2591, October 2006.


Published: October 2006
First available in Project Euclid: 23 January 2007

zbMATH: 1106.62029
MathSciNet: MR2291510
Digital Object Identifier: 10.1214/009053606000000821

Primary: 62C05 , 62F10 , 62F12 , 62J05 , 62J07

Keywords: Akaike’s information criterion AIC , consistency , Inference after model selection , lower risk bound , model uncertainty , post-model-selection estimator , pre-test estimator , selection of regressors , thresholding , uniform consistency

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 5 • October 2006
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