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August, 1969 Asymptotic Theory of a Class of Tests for Uniformity of a Circular Distribution
R. J. Beran
Ann. Math. Statist. 40(4): 1196-1206 (August, 1969). DOI: 10.1214/aoms/1177697496


Let $(x_1, x_2, \cdots, x_n)$ be independent realizations of a random variable taking values on a circle $C$ of unit circumference, and let $T_n = n^{-1} \int^1_0 \lbrack \sum^n_{j=1} f(x + x_j) - n \rbrack^2 dx,$ where $f(x)$ is a probability density on $C, f \varepsilon L_2\lbrack 0, 1 \rbrack$, and the addition $x + x_j$ is performed modulo 1. $T_n$ is used to test whether the observations are uniformly distributed on $C$. It includes as special cases several other statistics previously proposed for this purpose by Ajne, Rayleigh and Watson. The main results of the paper are the asymptotic distributions of $T_n$ under fixed alternatives to uniformity and under sequences of local alternatives to uniformity. A characterization is found for those alternatives against which $T_n$, with specified $f(x)$, gives a consistent test. The approximate Bahadur slope of $T_n$ is calculated from the asymptotic null distribution; however, an example indicates that this slope may not always reflect the power of $T_n$ reliably. A Monte Carlo simulation for a special case of $T_n$ suggests that a fair approximation to the power of $T_n$ may be obtained from its mean and variance under the alternative.


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R. J. Beran. "Asymptotic Theory of a Class of Tests for Uniformity of a Circular Distribution." Ann. Math. Statist. 40 (4) 1196 - 1206, August, 1969.


Published: August, 1969
First available in Project Euclid: 27 April 2007

zbMATH: 0211.51101
MathSciNet: MR261763
Digital Object Identifier: 10.1214/aoms/1177697496

Rights: Copyright © 1969 Institute of Mathematical Statistics

Vol.40 • No. 4 • August, 1969
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