The heart of a direct solution of the problem of moments of moments is a formula for the expected value of a product of sample power sums in terms of a linear function of products of population power sums. Such a result was given by Skellam (1949) and, earlier and more generally, by Dwyer (1938). The approach here leads to a result for the easily handled case which features the product of unit power sums of each multivariate variable from which a general case comes by easy extension without the need of technical material required by the earlier approaches. A corollary provides a new solution for the problem of obtaining the unbiased estimate of any linear function of products of power sums.
It is well-known that Wilks’ Λ criterion is distributed as the product of p independent beta variables in the p-variable null-case [3]. In the collinear case, Λ is still distributed as the product of p independent beta variables, one of them following a non-central beta density. Thus when p=2, the exact non-null distribution of Λ in the collinear case is given by the product of two independent beta variables, one central and the other having non-centrality parameter λ.
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