Asymptotic Theory of a Class of Tests for Uniformity of a Circular Distribution

Abstract

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.