On testing a class of restricted hypotheses

The aim of this paper is to provide the asymptotic distribution function, under the null and the alternative hypothesis, of statistics based on the likelihood, when the parameter space is restricted by a cone C such that DC is a spherical cone or the polyhedral cone {t∈RN+1:t1⩾0,…,tN+1⩾0} where D is a fixed known positive definite matrix. The results obtained ensure the calculation of threshold and power of such restricted tests. Our study is not limited to i.i.d. observations. Numerical calculations show that the likelihood ratio test restricted by a cone C1⊂C2 is uniformly more powerful than the likelihood ratio test restricted by C2 on all the alternatives that belong to C1. Particularly, a restricted test is uniformly more powerful than a non-restricted one on all the restricted alternatives.