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

SHINTO EGUCHI

Department of Mathematics, Shimane University, Matue 690, Japan

(Received October 24, 1989; revised February 22, 1990)

Abstract. 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.

Key words and phrases: Curvature tensor, composite null hypothesis, likelihood ratio, local power, local unbias, nuisance parameters, variance coefficient.