Some Variants of Chi-Square for Testing Uniformity on the Circle

One of the basic problems in the analysis of circularly distributed data is to find whether a given set of observations on the circumference of the unit circle indicate any preferred direction or whether the data can be considered to have come from a uniform distribution on the circumference. We shall assume, throughout this discussion, that the observations are given in terms of angles measured with respect to some suitably chosen origin (or zero direction), taking say, the anticlockwise direction as positive. A "goodness of fit" problem on the circle then is to test whether a random sample (e~, ..., c~,) comes from a population with a completely specified distribution function Fo(e), 0<e_<2n. If the specified distribution function is continuous, then the points x~ = F 0 (c~i) may be considered as observations on the circle of unit circumference, where now, the problem is to test whether the observations (x~, ..., x,) come from a uniform distribution. Thus a goodness of fit problem on the circle can also be reduced to testing for uniformity on the circle and the two problems are canonically equivalent just as they are on the line. Broadly speaking, the test procedures available for this purpose on the line may be grouped into three categories viz., (i) the methods based o n )~2 (ii) the methods utilising the empirical distribution functions and (iii) those based on sample spacings, i.e., differences between successive order statistics. However, these methods are not, in general, directly applicable for observations on the circle because of special problems posed by the arbitrary choice of the zero direction. A test statistic should, clearly be independent of this arbitrary origin, in order that it can be meaningfully used with the circular data. In some cases, modifications of the usual test statistics on the line, so as to make them independent of the choice of origin, have been introduced for use with the circular data. For instance, when employing the methods based on empirical distribution functions, Kuiper [61 and Watson [111 suggested such modifications for the standard Kolmogorov-Smirnov and Cramer-von Mises tests respectively. On the other hand, if the Z 2 methods were to be exploited for testing uniformity, one can make the usual Z 2 test for uniformity invariant under choice of origin, by considering the maximum possible value of Z 2 (for a given number of class intervals) or by taking the average such Z 2. We obtain the asymptotic distribution of the latter in Section 2 by using methods similar to those in Watson