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TESTING FOR HOMOGENEITY IN MIXTURE MODELS
Published online by Cambridge University Press: 24 July 2017
Abstract
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Statistical models of unobserved heterogeneity are typically formalized as mixtures of simple parametric models and interest naturally focuses on testing for homogeneity versus general mixture alternatives. Many tests of this type can be interpreted as C(α) tests, as in Neyman (1959), and shown to be locally asymptotically optimal. These C(α) tests will be contrasted with a new approach to likelihood ratio testing for general mixture models. The latter tests are based on estimation of general nonparametric mixing distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimator. Recent developments in convex optimization have dramatically improved upon earlier EM methods for computation of these estimators, and recent results on the large sample behavior of likelihood ratios involving such estimators yield a tractable form of asymptotic inference. Improvement in computation efficiency also facilitates the use of a bootstrap method to determine critical values that are shown to work better than the asymptotic critical values in finite samples. Consistency of the bootstrap procedure is also formally established. We compare performance of the two approaches identifying circumstances in which each is preferred.
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Footnotes
This research was partially supported by NSF grant SES-11-53548 and Project C1 of the SFB 823 of the German Research Foundation. Part of this research was conducted while the first author was visiting the Mathematics department at Ruhr University Bochum and the third author was a visiting scholar at UIUC. They are very grateful to the UIUC Statistics and Economics departments and the Bochum Mathematics department for their hospitality. The third author also gratefully acknowledges Financial support from the DFG (grant VO1799/1-1). The authors would also like to express their appreciation to the Editor, the Co-Editor and the referees for comments that led to improvements in the article.
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