The density of a quadratic form in a vector uniformly distributed on the n-sphere

There are many instances in the statistical literature in which inference i s based on a normalized quadratic form in a standard normal vector, normali zed by the squared length of that vector. Examples include both test statis tics (the Durbin-Watson statistic) and estimators (serial correlation coeff icients). Although much studied, no general closed-form expression for the density function of such a statistic is known, This paper gives general for mulae for the density in each open interval between the characteristic root s 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 multipli cities greater than one. Starting from a representation of the density as a surface integral over an (n - 2)-dimensional hyperplane, the density is ex pressed in terms of top-order zonal polynomials involving difference quotie nts of the characteristic roots of the matrix in the numerator quadratic fo rm.