The Annals of Statistics

Sensitive and Sturdy $p$-Values

John I. Marden

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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.

Article information

Ann. Statist., Volume 19, Number 2 (1991), 918-934.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F03: Hypothesis testing
Secondary: 62C20: Minimax procedures 62F04 62H15: Hypothesis testing 62C15: Admissibility

Hypothesis tests $p$-values robustness meta-analysis Fisher's procedure normal distribution spherical symmetry exponential family outliers Bartlett's problem


Marden, John I. Sensitive and Sturdy $p$-Values. Ann. Statist. 19 (1991), no. 2, 918--934. doi:10.1214/aos/1176348128.

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