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

by Gaffke, N.


Preprint series: 04-13, Preprints

62G10 Hypothesis testing
62G15 Tolerance and confidence regions


Abstract: 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.

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

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