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A note on profile likelihood for exponential tilt mixture models

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  • Z. Tan

Abstract

Suppose that independent observations are drawn from multiple distributions, each of which is a mixture of two component distributions such that their log density ratio satisfies a linear model with a slope parameter and an intercept parameter. Inference for such models has been studied using empirical likelihood, and mixed results have been obtained. The profile empirical likelihood of the slope and intercept has an irregularity at the null hypothesis so that the two component distributions are equal. We derive a profile empirical likelihood and maximum likelihood estimator of the slope alone, and obtain the usual asymptotic properties for the estimator and the likelihood ratio statistic regardless of the null. Furthermore, we show the maximum likelihood estimator of the slope and intercept jointly is consistent and asymptotically normal regardless of the null. At the null, the joint maximum likelihood estimator falls along a straight line through the origin with perfect correlation asymptotically to the first order. Copyright 2009, Oxford University Press.

Suggested Citation

  • Z. Tan, 2009. "A note on profile likelihood for exponential tilt mixture models," Biometrika, Biometrika Trust, vol. 96(1), pages 229-236.
    • Handle: RePEc:oup:biomet:v:96:y:2009:i:1:p:229-236
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    File URL: http://hdl.handle.net/10.1093/biomet/asn059
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    Cited by:

    1. Yang Ning & Yong Chen, 2015. "A Class of Pseudolikelihood Ratio Tests for Homogeneity in Exponential Tilt Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 504-517, June.
    2. Guanghua Han & Ming Dong, 2017. "Sustainable Regulation of Information Sharing with Electronic Data Interchange by a Trust-Embedded Contract," Sustainability, MDPI, vol. 9(6), pages 1-22, June.
    3. Jing Cheng & Jing Qin & Biao Zhang, 2009. "Semiparametric estimation and inference for distributional and general treatment effects," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(4), pages 881-904, September.
    4. Yufan Wang & Xingzhong Xu, 2023. "A Posterior p -Value for Homogeneity Testing of the Three-Sample Problem," Mathematics, MDPI, vol. 11(18), pages 1-25, September.
    5. Chuan Hong & Yang Ning & Shuang Wang & Hao Wu & Raymond J. Carroll & Yong Chen, 2017. "PLEMT: A Novel Pseudolikelihood-Based EM Test for Homogeneity in Generalized Exponential Tilt Mixture Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1393-1404, October.
    6. Pengfei Li & Yukun Liu & Jing Qin, 2017. "Semiparametric Inference in a Genetic Mixture Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1250-1260, July.

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