Distribution of a Quadratic Form in Normal Vectors

Summary

Let the column vectors of $\mathop {\text{X}}\limits_ \sim :$ pxn be distributed as independent normals with common covariance matrix $\mathop \sum \limits_ \sim$. Then, the quadratic form in normal vectors is denoted by $\mathop {\text{X}}\limits_ \sim \mathop {\text{A}}\limits_ \sim \mathop {\text{X}}\limits_ \sim '\, = \,\mathop {\text{S}}\limits_ \sim$ where $\mathop {\text{A}}\limits_ \sim :\,{\text{nxn}}$ is a symmetric matrix which is assumed to be positive definite. This paper deals with a series representation of the density function of $\mathop {\text{S}}\limits_ \sim$ when ${\text{E(}}\mathop {\text{X}}\limits_ \sim {\text{)}}\, \ne \,\mathop 0\limits_ \sim$, extending the idea of the author (1971) and Kotz et al. (1967b) to the multivariate non- central case. It is pointed out that this method gives a result which is not easy to obtain directly by integrating over an orthogonal space in the sense of James (1964).