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Nonparametric one-sided testing for the mean and related extremum problems.

by Gaffke, N..

Series: 2004-06, Preprints

62G10 Hypothesis testing
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)

We consider the nonparametric model of n i. i. d. nonnegative real random variables
whose distribution is unknown. An interesting parameter of that distribution is its
expectation mu. Wang & Zhao (2003) studied the problem of testing the one-sided
hypotheses H0 : mu <= mu_0 vs. H1: mu > mu_0 (with a given mu_0 > 0, where w.l.g. one may
take mu_0 = 1). For n = 1 there is a UMP nonrandomized level alpha test. Somewhat
surprisingly, for n = 2 Wang & Zhao obtained a UMP nonrandomized monotone
symmetric level alpha test. However, they conjectured that the result will not carry
over to larger sample size n >= 3. Unfortunately, their conjecture is true as we will
show. Also, we present an alternative proof of their (positive) result for n = 2.
Our derivations are based on a study of related classes of extremum problems on
products of probability measures.

Monotone symmetric test, UMP test, order statistics, probability measures, weak topology, semi-continuity.