# Distribution of a Quadratic Form in Normal Vectors

(Multivariate Non-Central Case)
• C. G. Khatri
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 17)

## 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).

## Key Words

Quadratic form in normal vectors series representation

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