On Elementary Symmetric Functions of the Roots of a Multivariate Matrix: Distributions

Abstract

Elementary symmetric functions (e.s.f.'s) of the roots of certain determinantal equations are often associated with test statistics in multivariate analysis. The $T^2$ of Hotelling (1951) is related to the first e.s.f. of the roots (this is also known as "sum of the roots" or "trace" of its associated matrix). The $\lambda$-criterion of Wilks (1932) appropriate to $k$ samples is related to the last e.s.f. of the roots (this is also known as "product of the roots"). Except for these two e.s.f.'s of the roots, very little is known in current literature of the other e.s.f.'s. Non-symmetric functions, like the largest or smallest root of Roy (1957), also arise in certain situations where their use are preferable to other known statistics. This paper is not concerned with symmetric or non-symmetric functions as test statistics. It is concerned with a unified treatment of the distribution problem of the e.s.f. of the roots. It begins with the derivation of the joint distribution of the e.s.f. of the roots of a multivariate matrix under null hypothesis (Section 2). The next three sections concern with the moment problem of the distribution. Section 3 cites previous material necessary for obtaining determinantal expressions of the moments. Section 4 deals with the proof by construction of obtaining determinantal expressions for moments and product-moments which is applied (Section 6) to the third moment of the second e.s.f. and the product-moments of the first, second and third e.s.f. for illustration purposes. Section 5 gives an evaluating formula for the determinantal expressions.