A GEOMETRIC LOOK AT NUISANCE PARAMETER EFFECT OF LOCAL POWERS IN TESTING HYPOTHESIS

This paper is concerned with the theory of testing hypothesis with composite null hypothesis or with nuisance parameters. The asymptotic behaviour of the likelihood ratio and the associated test statistics are investigated. Under a class of local alternatives with local orthogonality relative to the nuisance parameter vector, a unique decomposition of local power is presented. The decomposition consists of two parts; one is the influence of nuisance parameters and the other is the power corresponding to the simple case where the nuisance parameters are known. The decomposition formula is applied to some examples, including the gamma, Weibull and location-scale family.