Yuichi Takeda

James (1960) obtained the joint distributions of the characteristic roots of the covariance matrix on Wishart matrix by constituting the zonal polynomials. It was epoch-making in the history of multivariate distribution theory. Since that, the many density functions and moments in multivariate analysis have been expressed by the zonal polynomials and the generalized hypergeometric function of symmetric matrix argument. Examples are the noncentral distributions of the characteristic roots in multiple discriminant analysis (Constantine (1963)), the distributions of the largest characteristic root and the corresponding characteristic vector of a Wishart matrix (Sugiyama (1966,1967)) and the distributions of the largest and smallest root of a Wishart distribution of a multivariate beta distribution (Constantine (1963)). Sugiyama (1979) has obtained the coefficients of zonal polynomials up to degree 200 in the case of order 2 and the programming to compute it, expressed by a linear combination of monomial symmetric function. At the practical point of view, it is very important that the cumulative density function, moments, and so on are possible to compute in the case of order 3, and also higher order.

The distributions expressed by the generalized hypergeometric function may be written by the zonal polynomial series. Sugiyama, Fukuda and Takeda (1999) derived the partial differential equation to obtain the recurrence relations of coefficients of the generalized hypergeometric function for their computations. It is possible to calculate distribution expressed by generalized hypergeometric function order 7. However, at the present computers it is very difficult to calculate the coefficients of order more than 5. Hashiguchi and Niki (1997) derived the coefficients of zonal polynomials to elementary symmetric polynomials of order 3. They showed some results of the exact values of the percentile points of order 3.

In this paper, we discuss how to calculate zonal polynomials and generalized hypergeometric functions by using a partially differential equation. The densities of the largest characteristic roots are calculated numerically up to 5 dimensions. For the largest characteristic root, it might be impossible to calculate generalized hypergeometric function due to the infinite series. Therefore, we get truncated density function by a finite series and examine its precision. Some percentile points are also shown for these statistics.

**Keywords:** generalized hypergeometric function; zonal polynomials; largest latent root; principal component

**Biography:** I am associate Professor of Kanagawa Institute of Technology, Center for Basic Education and Integrated Learning, since Apr. 2006.

I worked for the Director (75th Anniversary) of Japan Statistical Society from June 2006 to June 2007, and also the Council menber since 2010.

Since 2011, I am the Director (Publications) of Japanese Socirety of Computational Statistical.