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Interpoint Distance Distributions and Their Applications Open Access Deposited

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The probability mass functions (p.m.f.) of the interpoint distance (IPD) between d-dimensional random vectors that are drawn from the multivariate power series distribution (MPSD) family and unified multivariate hypergeometric (UMHG) family are established. The distribution of IPD between multivariate normal vectors is obtained. The properties of IPDs within one sample and across two samples from distributions in these families are explored. The distribution of IPD between vectors that are drawn from a finite mixture of distributions within these families is acquired. The IPD distributions of members of the MPSD family are compared with each other and simulated distributions. We derive the p.m.f. of squared Euclidean norm and squared distance from fixed vectors in Z^d from distributions in the MPSD and UMHG families. The joint distribution of the MPSD and UMHG IPDs are determined and the distribution of order statistics of the conditional IPDs are also obtained. We discuss the applications of IPD for testing goodness of fit (GOF), the equality of distribution functions and bivariate exchangeability. Finite mixtures offer a rich class of distributions for modeling of a variety of random phenomena. Using the sample interpoint distances (IPDs), we propose the IPD-test for testing the hypothesis of homogeneity of the MPSD, UMHG and multivariate normal mixture models. The IPD-test is applied to mixture models for matrix-valued distributions and a test of homogeneity for Wishart mixture is presented. Numerical comparisons show that IPD-test has accurate type I errors and is more powerful in most multivariate cases than the EM-test and modified likelihood ratio test (MLRT).

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