## The Annals of Statistics

### Uniformly Powerful Goodness of Fit Tests

Andrew R. Barron

#### 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.

#### Article information

Source
Ann. Statist., Volume 17, Number 1 (1989), 107-124.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347005

Digital Object Identifier
doi:10.1214/aos/1176347005

Mathematical Reviews number (MathSciNet)
MR981439

Zentralblatt MATH identifier
0674.62032

JSTOR