###
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.

**Source**
( TeX ,
DVI ,
PS )