### 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(theta1,theta2) = 0 versus min(theta1,theta2) > 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.

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