The Number of Classes in Chi-Squared Goodness-of-Fit Tests

Abstract

The power of Pearson chi-squared and likelihood ratio goodness-of-fit tests based on different partitions is studied by considering families of densities “between” the null density and fixed alternative densities. For sample sizes n → ∞, local asymptotic theory with respect to the number of classes k is developed for such families. Simple sufficient and almost necessary conditions are derived under which it is asymptotically optimal to let k tend to infinity with n. A numerical study shows that the results of the asymptotic local theory for contamination families agree well with the actual power performance of the tests. For heavy-tailed alternatives, the tests have the best power when k is relatively large. Unbalanced partitions with some small classes in the tails perform surprisingly well, in particular when the alternatives have fairly heavy tails.

Key Words:

• Monte Carlo power
• Likelihood ratio test