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 ₘ䃑 ₘ₍ ₈=₁N f^1-zᵢ (-yᵢ) f^zᵢ (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.