## 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.

### MSC:

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