On Some Nuisance Parameter Free Uniformly Most Powerful Tests

TAKEUCHI (1969) provides a uniformly most powerful (UMP) one side test for testing the location parameter of the two parameters exponential model when the scale parameter is unknown. The power of his similar size α test depends, however, on the unknown scale parameter. In this case and in more general situations when there exists a sufficient statistic for the nuisance parameter, the theory of generalized THOMPSON's distributions, more specifically, the Thompsonization of a test statistic, LAURENT (1959, 1972) provides a UMP test whose power does not depend on the nuisance parameter. Examples of application of the general nuisance parameter free test procedure include here the truncated exponential, the inverse Gaussian, and the geometric distributions.