by Gaffke, N..

**Series:** 2004-13, Preprints

- MSC:
- 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.