A score test under a semiparametric finite mixture model

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We propose a score statistic to test the mixing proportion under a semiparametric finite mixture model introduced by Anderson [Anderson, J.A., 1979, Multivariate logistic compounds. Biometrika, 66, 17–26]. The proposed score test is based on the semiparametric profile log-likelihood function under a three-sample semiparametric model. The proposed score statistic has an asymptotic χ2 distribution under the null hypothesis and an asymptotic noncentral χ2 distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to the Wald test and the empirical likelihood ratio test. We present some results on simulation and on the analysis of one data set.