An Asymptotic Expansion for Permutation Tests with Several Samples

Abstract

Let $V_n$ be the standardized sum of squares of the means of $r + 1$ random samples of sizes $s_0, s_1, \cdots, s_r$, where $n = s_0 + s_1 + \cdots + s_r$, taken without replacement from $n$ numbers. Then using an approximation to the characteristic function of the means, an asymptotic expansion is obtained for the distribution of $V_n$ with first term being the distribution function of $\chi^2_r$ and with error of approximation generally of smaller order than $1/n$. When the numbers are the first $n$ integers, $V$ is the Kruskal-Wallis statistic and the approximation is compared with the exact distribution in some examples of this special case.