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Pearson goodness-of-fit test; Pearson-type goodness-of-fit tests; asymptotic local test power; asymptotic equivalence of tests; optimal number of classes
An 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.
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