Local score tests in mixture exponential family

Abstract

In this paper, we consider the testing problem for homogeneity in the mixture exponential density or probability distribution family f( x| θ, λ)=(1− λ) f 0( x)+ λf θ ( x), where f θ(x)=f 0(x) exp(t(x)θ−c(θ)) belongs to a one-parameter exponential family with c(0)=0. For testing H 0 : λθ=0 , we propose two score-based tests, called the supplementary score test and the separate score test, which only depend on the first two orthogonalized polynomial statistics. Their powers are compared with the ones of the generalized likelihood ratio test and other alternative tests for the normal and binomial mixture models. The numerical results show that the proposed tests are very competitive.