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Model selection for generalized linear models with factor-augmented predictors. (English) Zbl 1223.62129
The authors deal with generalized linear models in rich data environments. In particular, they consider the case where the sample size and the number of explanatory variables are of similar size. In such cases, dimension reduction is necessary for statistical inference. For dimension reduction the authors adopt the idea of principal components regression and assume that a small number of common factors of the explanatory variables are sufficient to describe the relevant information concerning the dependent variables. The common factors are latent variables and must be constructed from the observable explanatory variables. This study considers two important issues of generalized linear models with factor-augmented regressors. The first issue is the selection of the number of factors that best explain the response variables. The second issue is the selection of a distributional assumption for the response variables. Taking into account the effects of estimated regressors, the authors develop an information-theoretic criterion for model misspecifications for both the distributional and structural assumptions. Under the conditions $$T^{5/8}/N\to0$$ and $$\sqrt{N}/T\to0$$, it is shown that the bias term of the proposed estimate $$\min\{N,T\sqrt{T}\}$$ is consistent. The proposed criterion is a natural extension of the Akaike information criterion. Simulations and an empirical data analysis demonstrate that the proposed new criterion outperforms the Akaike information criterion and the Bayesian information criterion.

MSC:
 62J12 Generalized linear models (logistic models) 62H25 Factor analysis and principal components; correspondence analysis 62B10 Statistical aspects of information-theoretic topics 65C60 Computational problems in statistics (MSC2010)
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