Abstract
A general procedure for deriving asymptotic expansions for the distributions of multivariate statistics with symmetric property is given. To overcome the difficulty in obtaining higher order cumulants or moments, where huge amount of algebraic computation is required, we have developed an algorithm for the change of bases of multi-system symmetric polynomials and implemented it on computers as Lisp programs. Using them, together with the program library in REDUCE language composed for this research field by the authors, brings us expansion formulae expressed in terms of population moments or cumulants. An asymptotic expansion for the distribution of sample regressoin coefficient is illustrated as an example.
