SIMPLE EXACT BOUNDS FOR DISTRIBUTIONS OF LINEAR SIGNED RANK STATISTICS

Three types of simple bounds on the tail areas of arbitrary linear signed rank statistics are proposed: exponential (Bernstein-type) bounds, Chebyshev bounds and Berry-Esseen-Zolotarev bounds. All of them are valid for arbitrary sample size, explicit and readily computable, hence fully operational in testing or confidence estimation problems. No restriction whatever is made on score functions, which is of particular interest when 'regression constants' are involved, or when dealing with ties or discrete data. Unlike signed rank tests relying on approximate critical values derived from asymptotic normal distributions or asymptotic expansions, the tests based on the bounds proposed here are exact, in the sense that they never reject too often under the null hypothesis.