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Abstract:
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In choosing a candidate model in likelihood-based modeling via an information criterion, the practitioner is often faced with the difficult task of deciding just how far up the ranked list to look. Motivated by this pragmatic necessity, we construct an uncertainty band for a generalized (model selection) information criterion (GIC), defined as a criterion for which the limit in probability is identical to that of the normalized log-likelihood. This includes common special cases such as AIC & BIC. The method starts from the asymptotic normality of the GIC for the joint distribution of the candidate models in an independent and identically distributed (IID) data framework, and proceeds by deriving the (asymptotically) exact distribution of the minimum. This is a non-standard result from the theory of order statistics since although the original data are IID, the sample of GIC values are in fact dependent. The methodology is subsequently extended to two other commonly used model structures: regression and time series. Also, we explore ways in which the GIC uncertainty band can be inverted to make inferences about the parameters which means a post model selection inference.
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