Sensitive and sturdy \(p\)-values. (English) Zbl 0729.62016

Marden, John I.

Ann. Stat. 19, No. 2, 918-934 (1991).

Summary: We introduce new criteria for evaluating test statistics based on the \(p\)-values of the statistics. Given a set of test statistics, a good statistic is one which is robust in being reasonably sensitive to all departures from the null implied by that set.
We present a constructive approach to finding the optimal statistic. We apply the criteria to two-sided problems; combining independent tests; testing that the mean of a spherical normal distribution is 0, and extensions to other spherically symmetric and exponential distributions; Bartlett’s problem of testing the equality of several normal variances; and testing for one outlier in a normal linear model. For the most part, the optimal statistic is quite easy to use. Often, but not always, it is the likelihood ratio statistic.

MSC:

62F03	Parametric hypothesis testing
62F05	Asymptotic properties of parametric tests
62H15	Hypothesis testing in multivariate analysis

Keywords:

robustness; meta-analysis; Fisher’s procedure; spherical symmetry; \(p\)-values; two-sided problems; spherical normal distribution; exponential distributions; Bartlett’s problem; normal linear model; optimal statistic; likelihood ratio statistic