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Asymptotic theory of the likelihood ratio test for the identification of a mixture. (English) Zbl 1061.62028
Summary: The problems that arise when using the likelihood ratio test for the identification of a mixture distribution are well known: non-identifiability of the parameters and null hypotheses corresponding to a boundary point of the parameter space. In their approach to the problem of testing homogeneity against a mixture with two components, J. K. Ghosh and P. K. Sen [L. Le Cam and R. Olshen (eds.), Proc. Berkeley Conf. Honor J. Neyman and J. Kiefer, 789–806 (1985)] took into account these specific problems. Under general assumptions, they obtained the asymptotic distribution of the likelihood ratio test statistic. However, their result requires a separation condition which is not completely satisfactory. We show that it is possible to remove this condition with assumptions which involve the second derivatives of the density only.

MSC:
62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
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