Sensitive and Sturdy $p$-Values
The Annals of Statistics · 1991 · Volume 19 · Issue 2
Abstract
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.
Keywords
$P$-values, 62C15, 62C20, 62F03, 62F04, 62H15, Bartlett's problem, exponential family, Fisher's procedure, Hypothesis tests, Meta-analysis, normal distribution, Outliers, robustness, spherical symmetry
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