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Three test statistics for a nonparametric one-sided hypothesis on the mean of a nonnegative variable

by Gaffke, N..

Series: 2004-13, Preprints

62G10 Hypothesis testing
62G15 Tolerance and confidence regions

Assume the nonparametric model of $n$ i. i. d. nonnegative
real random variables whose distribution is unknown.
Consider the one sided hypotheses on the expectation,
$H_0 : \mu \leq 1$ vs. $H_1 : \mu > 1$. Wang & Zhao (2003)
studied several statistics for significance testing. Here
we focus on three statistics. One was introduced in Wang &
Zhao (2003), $W$ say, another is the nonparametric
likelihood ratio statistic $(R)$ also studied in that paper,
and last but not least we propose a new statistic $(K)$.
Either of these statistics has its values between zero and
one, and it seems reasonable to reject the null hypothesis
iff the value is smaller than or equal to $\alpha$ (the
nominal significance level). However, when doing so, the
question is whether the desired level $\alpha$ is really
kept. For $n \leq 2$ the answer is positive as shown by
Wang & Zhao (2003) for $W$ and $R$, and hence positive for
$K$ as well, since we will show that $W \leq K \leq R$
(for arbitray $n$). For $n \geq 3$ the answer is negative
for $W$ as shown by Gaffke (2004), but the definite answers
for $R$ and $K$ are unknown. We will report some numerical
evidence and an asymptotic result on the statistic $K$
which let us conjecture that the answer for $K$ (hence for
$R$ as well) is positive for arbitrary sample size. Some
what surprisingly, the numerics indicate that this should
be true even when we suspend the assumption of
{\it identically} distributed observations. For $n = 2$
this is proved.

Level of a test, UMP test, order statistics, stochastic ordering, asymptotic distribution, finite sample distribution.