The Annals of Mathematical Statistics

Limit Theorems Associated with Variants of the Von Mises Statistic

M. Rosenblatt

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A multidimensional analogue of the von Mises statistic is considered for the case of sampling from a multidimensional uniform distribution. The limiting distribution of the statistic is shown to be that of a weighted sum of independent chi-square random variables with one degree of freedom. The weights are the eigenvalues of a positive definite symmetric function. A modified statistic of the von Mises type useful in setting up a two sample test is shown to have the same limiting distribution under the null hypothesis (both samples come from the same population with a continuous distribution function) as that of the one-dimensional von Mises statistic. We call the statistics mentioned above von Mises statistics because they are modifications of the $\omega^2$ criterion considered by von Mises [5]. The paper makes use of elements of the theory of stochastic processes.

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Ann. Math. Statist., Volume 23, Number 4 (1952), 617-623.

First available in Project Euclid: 28 April 2007

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Rosenblatt, M. Limit Theorems Associated with Variants of the Von Mises Statistic. Ann. Math. Statist. 23 (1952), no. 4, 617--623. doi:10.1214/aoms/1177729341.

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