Contributions to the Theory of Rank Order Statistics--The One-Sample Case

Abstract

The one-sample problem is considered using techniques developed earlier [2], [3]. Let $Z = (Z_1, \cdots, Z_N)$ be a random vector with $Z_i = 1(0)$ if the $i$th smallest in absolute value in a sample of $N$ from the density $f(x)$ is positive (negative). Then $$P(Z = z) = N! \int_{\cdots_{0\leqq y_1\leqq\cdots\leqq yN\leqq\infty}}\int \prod_{i=1}^N \lbrack f^{1-z_i} (-y_i)f^{z_i}(y_i) dy_i\rbrack$$ Conditions are found implying $P(Z = z) > P(Z = z')$ where $z$ is derived from $z'$ by replacing a 0 by a 1, or interchanging a 0 and 1 in $z'$ by moving the 1 to the right. These conditions are met by the normal and other distributions. The results are useful in finding good tests of such null hypotheses as $X_1, \cdots, X_N$ are independently and identically distributed symmetrically about zero against such alternatives as slippage to the right. The Wilcoxon one sample signed rank test is a typical nonparametric procedure used under these conditions [4].