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

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 μ. Wang & Zhao (2003) studied the problem of testing the one-sided hypotheses H0 : μ ≤ μ0 vs. H1 : μ > μ0 (with a given μ0 > 0, where w.l.g. one may take μ0 = 1). For n = 1 there is a UMP nonrandomized level α test. Somewhat surprisingly, for n = 2 Wang & Zhao obtained a UMP nonrandomized monotone symmetric level α 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.