Uniformly Powerful Goodness of Fit Tests

Abstract

The simple hypothesis is tested that the distribution of independent random variables $X_1, X_2, \cdots, X_n$ is a given probability measure $P_0$. Let $\pi_n$ be any sequence of partitions. The alternative hypothesis is the set of probability measures $P$ with $\sum_{A\in \pi_n}|P(A) - P_0(A)| \geq \delta$, where $\delta > 0$. Note the dependence of this set of alternatives on the sample size. It is shown that if the effective cardinality of the partitions is of the same order as the sample size, then sequences of tests exist with uniformly exponentially small probabilities of error. Conversely, if the effective cardinality is of larger order than the sample size, then no such sequence of tests exists. The effective cardinality is the number of sets in the partition which exhaust all but a negligible portion of the probability under the null hypothesis.