Efficiencies of Chi-Square and Likelihood Ratio Goodness-of-Fit Tests

The classical problem of choice of number of classes in testing goodness of fit is considered for a class of alternatives, for the chi-square and likelihood ratio statistics. Pitman and Bahadur efficiencies are used to compare the two statistics and also to analyse the effect for each statistic of changing the number of classes for the case where the number of classes increases asymptotically with the number of observations. Overall, the results suggest that if the class of alternatives is suitably restricted the number of classes should not be very large.