Yasunori Fujikoshi

For discriminant analysis with (*q* + 1) *p*-dimensional populations, most inferential problems are based on two matrices *S*_{b} and *S*_{w} of sums of squares and products due to between-groups and within-groups, respectively. In this paper we are interesting in asymptotic behaviors of the coefficients on the linear discriminant functions, which are the characteristic vectors of *S*_{w}^{−1}*S*_{b}. Under normality *S*_{b} and *S*_{w} are independently distributed as Wishart distributions *W*_{p}(*q*,Σ; Ω) and *W*_{p}(*n* − *q* − 1, Σ), where *n* depends on the total sample size and Ω is the noncentrality matrix whose order is usually assumed to be *O*(*n*). Since the large-sample asymptotic distributions of the characteristic roots were derived by Hsu (1941), the results have been extended by obtaining their asymptotic expansions and treating the characteristic vectors. 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*.

Recently the distributions of the characteristic roots were studied in a high-dimensional situation where when the ratio *p*/*n* → *c* ∈ (0, 1) by Fujikoshi et al. (2008). In this paper we extended the high-dimensional asymptotic results to the case that Ω has any multiplicity, and study asymptotic behavior of the transformed characteristic vecors, which may be applied for statistical inference of the coefficients on the linear discriminant functions. We also discuss with similar problems in canonical correlation analysis. Some relationships between high-dimensional and large-sample approximations are pointed. Further, it is noted that the consistency properties of the sample roots and the vectors in large-sample case do not hold in high-dimensional case.

**Keywords:** High-dmensional approximations; Discriminant functions; Canonical correlation variates; Characteristic roots and vectors

**Biography:** The present position is an Emeritous Professor of Hiroshima University. I got Dr. Science from Hiroshima University in 1970. The title of thesis is “Asymptotic expansions of the distributions of test statistics in multivariate analysis. I am the author of the following books.

M. Siotani, T. Hayakawa and Y. Fujikoshi (1985). Modern Multivariate Statistical Analysis: A Graduate Course and Handbook. American Sciences Press, INC., Columbus, Ohio, U.S.A.

Y. Fujikoshi, V. V. Ulyanov and R. Shimizu (2010). Multivariate Statistics: High-Dimensional and Large-Sample Approximations. Wiley, Hoboken, New Jersey.