## The Annals of Statistics

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

### Sensitive and Sturdy $p$-Values

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176348128

**Digital Object Identifier**

doi:10.1214/aos/1176348128

**Mathematical Reviews number (MathSciNet)**

MR1105852

**Zentralblatt MATH identifier**

0729.62016

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F03: Hypothesis testing

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

**Keywords**

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

#### Citation

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