The Annals of Mathematical Statistics

Limit Theorems Associated with Variants of the Von Mises Statistic

M. Rosenblatt

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177729341

Digital Object Identifier
doi:10.1214/aoms/1177729341

Mathematical Reviews number (MathSciNet)
MR52732

Zentralblatt MATH identifier
0048.36003

JSTOR
links.jstor.org

Citation

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. https://projecteuclid.org/euclid.aoms/1177729341


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