Abstract:
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Let X_1, ..., X_n be independent observations with X_i~N(\theta_i,\tau_i^2), where means and variances are unknown and \sigma>0 is a known lower bound for the unknown variances. Let f_n be the average marginal density of observations. We consider the problem of testing if f_n belongs to a family of location-scale mixture densities. We study a generalized likelihood ratio test (GLRT) based on the generalized maximum likelihood estimator (GMLE, Robbins, 1950; Kiefer and Wolfowitz, 1956). By an equivalence between the heteroscedastic and homoscedastic Gaussian models, we establish a large deviation inequality of the GLRT under the null hypothesis. The inequality gives an upper bound of the significance level of the test. We provide a sufficient condition under which the GLRT has asymptotically full power. For the two-component heterogeneous and heteroscedastic Gaussian mixture, it turns out that the GLRT has asymptotically full power throughout the entire detectable region. We demonstrate the power of the GLRT for moderate samples with numerical experiments.
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