P-VAlUES MAXIMIZED OVER A CONFIDENCE SET FOR THE NUISANCE PARAMETER

For testing problems of the form Ho : v = Vo with unknown nuisance parameter (). a variety of methods are used to deal with (). The simplest approach is exemplified by the t test where the unknown variance is replaced by the sample variance. and the t distribution accounts for estimation of the variance. In other problems such as the two by two contingency table. one conditions on a sufficient statistic for () and proceeds as in Fisher's exact test. Since neither of these standard methods is appropriate for all situations, this paper suggests a new method for handling the unknown' (). This new method is a simple modification of the formal definition of a p-value which involves taking a maximum over the nuisance parameter space of a p-value obtained for the case when () is known. The suggested modification is to restrict the maximization to a confidence set for the nuisance parameter. After giving a briefjustification, a variety of examples show how this new method gives improved results for two by two tables and solves previously intractable semi-parametric problems. The library 0·' the Depertment of Statistics North Carolina State University