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

✍ Scribed by Hillier, Grant

• Publisher: Cambridge University Press
• Year: 2001
• Tongue: English
• Weight: 164 KB
• Volume: 17
• Category: Article
• ISSN: 0266-4666
• DOI: 10.1017/s026646660117101x

✦ Synopsis

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.