Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T01:08:30.418Z Has data issue: false hasContentIssue false

THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE

Published online by Cambridge University Press:  07 February 2001

Grant Hillier
Affiliation:
University of Southampton

Abstract

There are many instances in the statistical literature in which inference is based on a normalized quadratic form in a standard normal vector, normalized by the squared length of that vector. Examples include both test statistics (the Durbin–Watson statistic) and estimators (serial correlation coefficients). Although much studied, no general closed-form expression for the density function of such a statistic is known. This paper gives general formulae for the density in each open interval between the characteristic roots of the matrix involved. Results are given for the case of distinct roots, which need not be assumed positive, and when the roots occur with multiplicities greater than one. Starting from a representation of the density as a surface integral over an (n − 2)-dimensional hyperplane, the density is expressed in terms of top-order zonal polynomials involving difference quotients of the characteristic roots of the matrix in the numerator quadratic form.

Type
Research Article
Copyright
© 2001 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)