An Exact Decomposition Theorem for a Sample from the Three-Dimensional Hyperboloid Distribution
该文证明三维双曲面分布样本的联合概率密度可分解为n个独立指数κ分布的乘积，并给出类似结果在正态、逆高斯和伽马分布上的应用，适用于递归检验。
SUMMARY Consider a sample (v 1, ..., vn) from the three-dimensional hyperboloid distribution. The joint probability distribution function (PDF) of (v 1 ..., vn) can be expressed as the product of the PDF of n independent quantities. Each of these quantities has an exponential κ distribution. The splitting is done in such a way that the data points vi are incorporated into the successive quantities one at a time. A similar result has been proved in the literature for the normal and the inverse Gaussian distributions. Since the normal, gamma and inverse Gaussian distributions are the only one-dimensional distributions which are reproductive and are equal to their saddlepoint approximation, a parallel result was to be expected for the gamma distribution. We also prove this result and an application to recursive testing is presented.