# The small-sample power of some nonparametric tests

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I. Small-Sample Power of the One-Sample Sign Test for Approximately Normal Distributions. The power function of the one-sided, one-sample sign test is studied for populations which deviate from exact normality, either by skewness, kurtosis, or both. The terms of the Edgeworth asymptotic expansion of order more than N^{-3/2} are used to represent the population density. Three sets of hypotheses and alternatives, concerning the location of (1) the median, (2) the median as approximated by the mean and coefficient of skewness, and (3) the mean, are considered in an attempt to make valid comparisons between the power of the sign test and Student's t test under the same conditions. Numerical results are given for samples of size 10, significance level .05, and for several combinations of the coefficients of skewness and kurtosis.

II. Power of Two-Sample Rank Teats on the Equality of Two Distribution Functions. A comparative study is made of the power of two-sample rank tests of the hypothesis that both samples are drawn from the same population. The general alternative is that the variables from one population are stochastically larger than the variables from the other.

One of the alternatives considered is that the variables in the first sample are distributed as the smallest of k variates with distribution F, and the variables in the second sample are distributed as the largest of these k – H₁ : H = 1 - (1 - F)^{k}, G = F^{k}. These two alternative distributions are mutually symmetric if F is symmetrical. Formulae are presented, which are independent of F, for the evaluation of the probability under H₁ of any joint arrangement of the variables from the two samples. A theorem is proved concerning the equality of the probabilities of certain pairs of orderings under assumptions of mutually symmetric populations. The other alternative is that both samples are normally distributed with the same variance but different means, the standardized difference between the two extreme distributions in the first alternative corresponding to the difference between the means. Numerical results of power are tabulated for small sample sizes, k = 2, 3 and 4, significance levels .01, .05 and .10. The rank tests considered are the most powerful rank test, the one and two-sided Wilcoxon tests, Terry's c₁ test, the one and two-aided median tests, the Wald-Wolfowitz runs test, and two new tests called the Psi test and the Gamma test.

The two-sample rank test which is locally most powerful against any alternative·expressing an arbitrary functional relationship between the two population distribution functions and an unspecified parameter θ is derived and its asymptotic properties studied. The method is applied to two specific functional alternatives, H₁* : H = (1-θ)F^{k} + θ[1 - (1-F)^{k}], G = F^{k}, and H₁**: H = 1 - (1-F)^{1+θ}, G = F^{1+θ}, where θ ≥ 0, which are similar to the alternative of two extreme distributions. The resulting test statistics are the Gamma test and the Psi test, respectively. The latter test is shown to have desirable small-sample properties.

The asymptotic power functions of the Wilcoxon and WaldWolfowitz tests are compared for the alternative of two extreme distributions with k = 2, equal sample sizes and significance level .05.