Asymptotic results in canonical discriminant analysis when the dimension is large compared to the sample size

This paper deals with the distributions of test statistics for the number of useful discriminant functions and the characteristic roots in canonical discriminant analysis. These asymptotic distributions have been extensively studied when the number p   of variables is fixed, the number q+1q+1 of groups is fixed, and the sample size N tends to infinity. However, these approximations become increasingly inaccurate as the value of p increases for a fixed value of N. On the other hand, we encounter to analyze high-dimensional data such that p is large compared to n. The purpose of the present paper is to derive asymptotic distributions of these statistics in a high-dimensional framework such that q   is fixed, p→∞p→∞, m=n-p+q→∞m=n-p+q→∞, and p/n→c∈(0,1)p/n→c∈(0,1), where n=N-q-1n=N-q-1. Numerical simulation revealed that our new asymptotic approximations are more accurate than the classical asymptotic approximations in a considerably wide range of (n,p,q)(n,p,q).