Watson's u_n statistic for a discrete distribution

A general method is given for obtaining approximate percentiles of the asymptotic distribution of Watson's U_N² for testing goodness of fit to a completely specified discrete distribution. The method is applied to a particular distribution arising in data on seasonal incidence. Monte Carlo simulations give estimates of percentiles which are generally consistent with those obtained by the approximate method. The asymptotic results approximate well for sample sizes as small as 25. The power of U_N² is estimated and compared to that of Kuiper's V_N and Edwards's test.