Criteria for selection of response variables and the asymptotic properties in a multivariate calibration

Let a set of p responses y=(y ₁,... y _p )′ has a multivariate linear regression on a set of q explanatory variables x=(x ₁,... x _q )′. Our aim is to select the most informative subset of responses for making inferences about an unknown x from an observed y. Under normality on y, two selection methods, based on the asymptotic mean squared error and on the Akaike's information criterion, are proposed by Fujikoshi and Nishii (1986, Hiroshima Math. J., 16, 269-277). In this paper, under a mild condition we will derive the cross-validation criterion and obtain the asymptotic properties of the three procedures.