On the density of generalised quadratic forms with applications to asymptotic expansions for test statistics

In this note we derive a general formula useful to express the density of generalised noncentral quadratic forms (i.e. of a scalar random variable obtained by contracting non zero mean multivariate normal vectors over multidimensional arrays) in terms of linear combinations of noncentral chi square random variables. The formula can be used to obtain explicit expressions for the terms appearing in the asymptotic expansions for test statistics under a local alternative.