THE POWER OF SOME TESTS FOR UNIFORMITY OF A CIRCULAR DISTRIBUTION.
JOHNS HOPKINS UNIV BALTIMORE MD DEPT OF STATISTICS
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Let x sub 1, x sub 2,...,x sub n be independent observations on an arbitrary random variable which takes values on a circle of unit circumference. Suppose fx is a probability density in L sub 20,1. The main result of this paper is the asymptotic distribution of the class of test statistics T sub n 1n multiplied by the integral taken between the limits 0,1 of the quantity summation from j 1 to j n of fx x sub j-n squared dx. T sub n is used to test whether the observations are uniformly distributed on the circle. It includes as special cases several other statistics previously proposed for this purpose by Rayleigh, Watson, Ajne and others. Two distinct cases arise in the asymptotics -- for one class of alternatives T sub n yields a consistent test for uniformity, but for the others T sub n gives a test which is not consistent. A fair approximation to the power of T sub n can be obtained from its first two moments. The approximate Bahadur slope of T sub n is calculated from its asymptotic null distribution it does not appear to reflect the power of T sub n reliably. Author
- Statistics and Probability