(Received August 7, 1989; revised October 25, 1990)
Abstract. When a testing problem has nuisance parameters, the uniformly most powerful (UMP) tests do not generally exist. Exceptional examples were given by Dubey (1962, Skand. Aktuarietidskr., 45, 25-38; 1963, Skand. Aktuarietidskr., 46, 1-24) and Takeuchi (1968, Ann. Math. Statist., 40, 1838-1839) for the exponential distributions. What is essential for proving the existence of UMP tests lies in a special relationship between null hypothesis and the alternative. Assuming a similar relationship between them, a similar kind of result can be shown under more general situation.
Key words and phrases: Uniformly most powerful test, nuisance parameter, exponential distribution, uniform distribution.
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