Strong consistency of the information criterion for model selection in multivariate analysis (Q1124251)

We consider a generalized information criterion (GIC) obtained by the information theoretic approach. According to this procedure, we find the model which minimizes \[ (1)\quad GIC=-2 \log L({\hat \theta})+pc_ N, \] where L(\({\hat \theta}\)) is the maximized likelihood and p is the number of parameters. Recently \textit{L. C. Zhao} and the last two authors [J. Multivariate Anal. 20, 1-25 (1986; Zbl 0617.62055)] considered the GIC such that \[ (2)\quad \lim_{N\to \infty}c_ N/N=0\quad and\quad \lim_{N\to \infty}c_ N/\log \log N=+\infty. \] The above criterion is sometimes referred to as the efficient detection (ED) criterion. They used the criterion for the determination of the number of signals under a signal processing model. In the present paper, we propose to use the ED criterion for certain problems of multivariate analysis. Sometimes the statistician is expected to predict the explanatory variables using some of the response variables under the multivariate regression model. This problem is treated in Section 2 by using the ED criterion, and its consistency is established. In Section 3 we discuss the selection of variables in discriminant analysis. Our interest is to find the variables which contribute to discrimination between the populations. Section 4 is concerned with canonical correlation analysis, i.e., among two sets of variables we want to find which subsets are important for studying the association between the two sets. The investigations for the above cases will be carried out under a mild condition on the underlying distribution.