The number of classes in chi-squared goodness-of-fit tests. (English) Zbl 0582.62037

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 \(n\to \infty\), 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.


62G10 Nonparametric hypothesis testing
65C05 Monte Carlo methods
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