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December, 1952 Limit Theorems Associated with Variants of the Von Mises Statistic
M. Rosenblatt
Ann. Math. Statist. 23(4): 617-623 (December, 1952). DOI: 10.1214/aoms/1177729341


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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M. Rosenblatt. "Limit Theorems Associated with Variants of the Von Mises Statistic." Ann. Math. Statist. 23 (4) 617 - 623, December, 1952.


Published: December, 1952
First available in Project Euclid: 28 April 2007

zbMATH: 0048.36003
MathSciNet: MR52732
Digital Object Identifier: 10.1214/aoms/1177729341

Rights: Copyright © 1952 Institute of Mathematical Statistics

Vol.23 • No. 4 • December, 1952
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