THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE
给出了标准正态向量归一化二次型密度函数的一般公式，适用于矩阵特征根互异或存在重根的情形，对Durbin-Watson统计量等检验统计量和序列相关系数估计量有直接应用。
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.