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

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.


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