An Asymptotic Expansion for the Distribution of the Linear Discriminant Function

Abstract

The distribution of the linear discriminant function $W$, Anderson's classification statistic (1951), is investigated by several authors: Bowker (1960), Bowker and Sitgreaves (1961), Sitgreaves (1952, 1961), etc. Since the exact distribution is too complicated to be used numerically, as indicated by Sitgreaves (1961), we present here an asymptotic expansion of the distribution with respect to three numbers $N_1, N_2$ and $n$ representing degrees of freedom. This is a generalization of the result of Bowker and Sitgreaves who deal with a special case where $N_1 = N_2 = N$ and $n = 2N - 2$.