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

The one-sample problem is considered using techniques developed earlier [2], [3]. Let Z = (Z₁, ⋯, ZN) be a random vector with Zᵢ = 1(0) if the ith smallest in absolute value in a sample of N from the density $f(x)$ is positive (negative). Then P(Z = z) = N! ∫_⋯0 y₁⋯ yN∞∫ ∏ᵢ₌₁N f1-zᵢ (-yᵢ)fzᵢ(yᵢ) dyᵢ 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 X₁, ⋯, XN 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].