Criteria for Linear Model Selection Based on Kullback's Symmetric Divergence

Authors


  • Acknowledgments. The author thanks the editor, the associate editor and two referees whose thoughtful suggestions helped to improve the original version of this manuscript. This research was supported by the National Science Foundation, grant DMS–9704436.

Summary

Model selection criteria are frequently developed by constructing estimators of discrepancy measures that assess the disparity between the 'true' model and a fitted approximating model. The Akaike information criterion (AIC) and its variants result from utilizing Kullback's directed divergence as the targeted discrepancy. The directed divergence is an asymmetric measure of separation between two statistical models, meaning that an alternative directed divergence can be obtained by reversing the roles of the two models in the definition of the measure. The sum of the two directed divergences is Kullback's symmetric divergence.

In the framework of linear models, a comparison of the two directed divergences reveals an important distinction between the measures. When used to evaluate fitted approximating models that are improperly specified, the directed divergence which serves as the basis for AIC is more sensitive towards detecting overfitted models, whereas its counterpart is more sensitive towards detecting underfitted models. Since the symmetric divergence combines the information in both measures, it functions as a gauge of model disparity which is arguably more balanced than either of its individual components. With this motivation, the paper proposes a new class of criteria for linear model selection based on targeting the symmetric divergence. The criteria can be regarded as analogues of AIC and two of its variants: 'corrected' AIC or AICc and 'modified' AIC or MAIC. The paper examines the selection tendencies of the new criteria in a simulation study and the results indicate that they perform favourably when compared to their AIC analogues.

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