On the optimal number of classes in the Pearson goodness-of-fit tests. (English) Zbl 1245.62045

Summary: The asymptotic local power of Pearson chi-squared tests is considered, based on convex mixtures of the null densities with fixed alternative densities when the mixtures tend to the null densities for sample sizes \(n\rightarrow \infty \). This local power is used to compare the tests with fixed partitions \(\mathcal {P}\) of the observation space of small partition sizes \(| \mathcal {P}|\) with the tests whose partitions \(\mathcal {P}=\mathcal {P}_n\) depend on \(n\) and the partition sizes \(| \mathcal {P}_n|\) tend to infinity for \(n\rightarrow \infty\). New conditions are presented under which it is asymptotically optimal to let \(| \mathcal {P}|\) tend to infinity with \(n\) or to keep it fixed, respectively. Similar conditions are presented under which the tests with fixed \(| \mathcal {P}|\) and those with increasing \(| \mathcal {P}_n|\) are asymptotically equivalent.


62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
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