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On Model Selection Curves

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  • Samuel Müller
  • Alan H. Welsh

Abstract

Many popular methods of model selection involve minimizing a penalized function of the data (such as the maximized log‐likelihood or the residual sum of squares) over a set of models. The penalty in the criterion function is controlled by a penalty multiplier λ which determines the properties of the procedure. In this paper, we first review model selection criteria of the simple form “Loss + Penalty” and then propose studying such model selection criteria as functions of the penalty multiplier. This approach can be interpreted as exploring the stability of model selection criteria through what we call model selection curves. It leads to new insights into model selection and new proposals on how to select models. We use the bootstrap to enhance the basic model selection curve and develop convenient numerical and graphical summaries of the results. The methodology is illustrated on two data sets and supported by a small simulation. We show that the new methodology can outperform methods such as AIC and BIC which correspond to single points on a model selection curve. Beaucoup de méthodes populaires de sélection de variables impliquent la minimisation d'une fonction pénalisée des données (comme la vraisemblance maximisée ou la somme résiduelle carrés) sur un jeu de modèles. La pénalité dans la fonction de critère est contrôlée par un multiplicateur de pénalitéλ qui détermine les propriétés de la procédure. Nous reconsidérons d'abord des critères de sélection modèles de la forme simple ‘Perte + Pénalité’ et proposons ensuite d'étudier de telles fonctions comme les fonctions du multiplicateur de pénalité. Cette approche peut être interprétée comme l'exploration de la stabilité de fonctions de critère par ce que nous appelons des courbes de choix modèles. Il mène à de nouvelles compréhensions dans le sélection de variables et de nouvelles propositions de la façon d'utiliser ces fonctions de critère pour sélectionner de variables. Nous utilisons le bootstrap pour augmenter des courbes de choix modèle et développent les résumés numériques et graphiques des résultats. La méthodologie est illustrée sur deux jeux de données et soutenue par une petite simulation. Nous montrons que la nouvelle méthodologie peut surpasser des méthodes comme AIC et BIC qui correspond aux points simples sur une courbe de choix modèle.

Suggested Citation

  • Samuel Müller & Alan H. Welsh, 2010. "On Model Selection Curves," International Statistical Review, International Statistical Institute, vol. 78(2), pages 240-256, August.
  • Handle: RePEc:bla:istatr:v:78:y:2010:i:2:p:240-256
    DOI: 10.1111/j.1751-5823.2010.00108.x
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    References listed on IDEAS

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    1. Yuhong Yang, 2005. "Can the strengths of AIC and BIC be shared? A conflict between model indentification and regression estimation," Biometrika, Biometrika Trust, vol. 92(4), pages 937-950, December.
    2. Claeskens,Gerda & Hjort,Nils Lid, 2008. "Model Selection and Model Averaging," Cambridge Books, Cambridge University Press, number 9780521852258, October.
    3. Zhao, L. C. & Krishnaiah, P. R. & Bai, Z. D., 1986. "On detection of the number of signals when the noise covariance matrix is arbitrary," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 26-49, October.
    4. Wisnowski, James W. & Simpson, James R. & Montgomery, Douglas C. & Runger, George C., 2003. "Resampling methods for variable selection in robust regression," Computational Statistics & Data Analysis, Elsevier, vol. 43(3), pages 341-355, July.
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    1. Jiming Jiang & Thuan Nguyen & J. Sunil Rao, 2015. "The E-MS Algorithm: Model Selection With Incomplete Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 1136-1147, September.

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