Conditions for Consistency of a Log-Likelihood-Based Information Criterion in Normal Multivariate Linear Regression Models under the Violation of the Normality Assumption

Abstract

In this paper, we clarify conditions for consistency of a log-likelihood-based information criterion in multivariate linear regression models with a normality assumption. Although normality is assumed for the distribution of the candidate model, we frame the situation so that the assumption of normality may be violated. The conditions for consistency are derived from two types of asymptotic theory; one is based on a large-sample asymptotic framework in which only the sample size approaches ∞, and the other is based on a high-dimensional asymptotic framework in which the sample size and the dimension of the vector of response variables simultaneously approach infinity. In both cases, our results are independent of any indicator measuring a discrepancy between the true distribution and the normal distribution, e.g., skewness, kurtosis and other higher-order cumulants of the true distribution.

Asymptotic Optimality of Cp-Type Criteria in High-Dimensional Multivariate Linear Regression Models

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