Hirofumi Wakaki

A consistency property of the AIC for multivariate linear models when the dimension and the sample size are large

It is common knowledge that the Akaike’s information criterion (AIC) is not a consistent model selection criterion. This inconsistency property has been confirmed from an asymptotic selection probability evaluated from a large-sample asymptotic framework. However, when a high-dimensional asymptotic framework, such that the dimension of the response variables and the sample size are approaching 1, is used for evaluating the selection probability, we can prove a consistency property of the AIC for selecting variables in multivariate linear models. This means that the probability of selecting the true model by the AIC goes to 1 as the sample size and the dimension simultaneously approach 1. The consistency property is also checked numerically by conducting a Monte Carlo simulation.

Bias Corrections of some Criteria for Selecting Multivariate Linear Models in a General Nonnormal Case

SYNOPTIC ABSTRACT This paper deals with the bias corrections of two types of information criteria for selecting multivariate linear regression models in a general nonnormal case. One type is the AIC-type criterion and the other is the Cp-type criterion. One of our goals is to study the effect of nonnormality in the biases of three criteria, i.e., AIC, TIC and Cp, by deriving their asymptotic expansions. From these results, new criteria having a smaller influence of nonnormality than these three criteria are proposed by using the bootstrap method. Biases of our criteria are always corrected to O(n−1). We point out some tendencies of these criteria and verify that our criteria are better than the three criteria by conducting numerical experiments.

High-dimensional Edgeworth expansion of a test statistic on independence and its error bound

In this paper, we calculate Edgeworth expansion of a test statistic on independence when some of the parameters are large, and simulate the goodness of fit of its approximation. We also calculate an error bound for Edgeworth expansion. Some tables of the error bound are given, which show that the derived bound is sufficiently small for practical use.

Optimal discriminant functions for normal populations

A class of discriminant rules which includes Fishers linear discriminant function and the likelihood ratio criterion is defined. Using asymptotic expansions of the distributions of the discriminant functions in this class, we derive a formula for cut-off points which satisfy some conditions on misclassification probabilities, and derive the optimal rules for some criteria. Some numerical experiments are carried out to examine the performance of the optimal rules for finite numbers of samples.