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ADMISSIBILITY OF UNBIASED TESTS FOR A COMPOSITE

HYPOTHESIS WITH A RESTRICTED ALTERNATIVE

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MANABU IWASA

*Department of Mathematical Science, Osaka University, Toyonaka, Osaka 560, Japan*
(Received November 29, 1989; revised November 1, 1990)

**Abstract.**
This paper discusses *alpha*-admissibility and
*d*-admissibility which are
important concepts in studying the performance of statistical tests for
composite hypotheses. A sufficient condition for *alpha*-admissibility is
presented. When *alpha* = 1/*m*, the Nomakuchi-Sakata test, which is
uniformly more powerful than the likelihood ratio test for hypotheses
min(*theta*_{1},*theta*_{2}) = 0 versus min(*theta*_{1},*theta*_{2}) > 0,
is generalized for a class of distributions in an exponential family, and
its unbiasedness and *alpha*-admissibility are shown. Finally, the case of
*alpha* \neq 1/*m* is discussed in brief.

*Key words and phrases*:
Nomakuchi-Sakata test,
*alpha*-admissibility, *d*-admissibility, unbiasedness,
exponential family, completeness.

**Source**
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