A NOTE ON THE UNIFORMLY MOST POWERFUL TESTS IN THE PRESENCE OF NUISANCE PARAMETERS

KENTARO NOMAKUCHI

Department of Mathematics, Kyushu University, Higashi-ku, Fukuoka 812, Japan

(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.