Bahadur Efficiencies of Some Tests for Uniformity on the Circle

Abstract

In this paper we compare the asymptotic efficiencies of several tests that are available for testing uniformity on the circle. Since the problem of testing goodness of fit on the circle can be reduced to testing uniformity by a simple probability transformation, these comparisons are applicable also to the goodness of fit situation. The alternatives to uniformity considered here are the familiar circular normal distributions (CND's) with density \begin{equation*}\tag{1.1}g(\alpha) = \lbrack 2\pi I_0(\kappa)\rbrack^{-1} \exp\lbrack\kappa \cos \alpha\rbrack,\quad - \pi \leqq \alpha < \pi.\end{equation*} $0 \leqq \kappa < \infty$ is a parameter of concentration, larger values of $\kappa$ corresponding to more concentration towards the mean direction zero, and $I_0(\kappa)$ is the Bessel function of purely imaginary argument. When $\kappa = 0 (1.1)$ is the uniform density, so the null hypothesis is $H_0: \kappa = 0$. The test compared here are (i) Ajne's test A (ii) Watson's test W (iii) Rayleigh's test R (iv) Ajne's test N (v) Kuiper's test V (iv) Spacings test U. In subsequent sections each of these tests is briefly described and its Bahadur efficiency [4], [5] is computed, using large deviation results. We compare the local slopes of the test statistics, i.e. the slopes in the neighborhood of the hypothesis. On the basis of these comparisons, we fine that limiting efficiencies of the first three tests viz., Ajne's test A, Watson's W and Rayleigh's test based on R, are identical, while the other tests have lower asymptotic efficiencies. Further conclusions are given in Section 7. Finally in Section 8 a simple inequality between the Ajne's N and Kuiper's V, whose asymptotic performances are identical, is noted.