## The Annals of Mathematical Statistics

### Small Sample Power of the One Sample Wilcoxon Test for Non-Normal Shift Alternatives

Harvey J. Arnold

#### Abstract

The power of the one sample Wilcoxon test is computed for the hypothesis that the median is zero against various shift alternatives for samples drawn from several different non-normal distributions. A recursive scheme given by Klotz [2] simplifies the problem of power computations and allows investigating samples to size $n = 10$ on a large computer. The power for selected type I errors $\alpha$ are compared with the power of a best signed-rank procedure obtained by sorting the probabilities in decreasing order for all possible sample configurations for fixed $n$ and adding up the probabilities associated with the most probable 100 $\alpha$% of the configurations. The non-normal distributions selected for study are the $t$ distribution with degrees of freedom $\frac{1}{2}$, 1, 2, and 4. The one sample Wilcoxon test, found to be powerful for normal shift alternatives by Klotz [2], deteriorates badly in power for the long-tailed distributions studied as does the one sample $t$ test. However, the Wilcoxon test remains more powerful than the $t$ test. The sign test is still more powerful than either Wilcoxon or $t$. No asymptotic results are obtained.

#### Article information

Source
Ann. Math. Statist., Volume 36, Number 6 (1965), 1767-1778.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177699805

Digital Object Identifier
doi:10.1214/aoms/1177699805

Mathematical Reviews number (MathSciNet)
MR183077

Zentralblatt MATH identifier
0136.40404

JSTOR