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On empirical likelihood for a semiparametric mixture model

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  • F. Zou

Abstract

Plant and animal studies of quantitative trait loci provide data which arise from mixtures of distributions with known mixing proportions. Previous approaches to estimation involve modelling the distributions parametrically. We propose a semiparametric alternative which assumes that the log ratio of the component densities satisfies a linear model, with the baseline density unspecified. It is demonstrated that a constrained empirical likelihood has an irregularity under the null hypothesis that the two densities are equal. A factorisation of the likelihood suggests a partial empirical likelihood which permits unconstrained estimation of the parameters, and which is shown to give consistent and asymptotically normal estimators, regardless of the null. The asymptotic null distribution of the log partial likelihood ratio is chi-squared. Theoretical calculations show that the procedure may be as efficient as the full empirical likelihood in the regular set-up. The usefulness of the robust methodology is illustrated with a rat study of breast cancer resistance genes. Copyright Biometrika Trust 2002, Oxford University Press.

Suggested Citation

  • F. Zou, 2002. "On empirical likelihood for a semiparametric mixture model," Biometrika, Biometrika Trust, vol. 89(1), pages 61-75, March.
    • Handle: RePEc:oup:biomet:v:89:y:2002:i:1:p:61-75
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    Citations

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    Cited by:

    1. Jing Qin & Denis H. Y. Leung, 2004. "A Semi-parametric Two-component “Compound” Mixture Model and Its Application to Estimating Malaria Attributable Fractions," Working Papers 17-2004, Singapore Management University, School of Economics.
    2. Moming Li & Guoqing Diao & Jing Qin, 2020. "On symmetric semiparametric two‐sample problem," Biometrics, The International Biometric Society, vol. 76(4), pages 1216-1228, December.
    3. Wang, Dong & Chen, Song Xi, 2009. "Combining quantitative trait loci analyses and microarray data: An empirical likelihood approach," Computational Statistics & Data Analysis, Elsevier, vol. 53(5), pages 1661-1673, March.
    4. Liugen Xue, 2010. "Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 644-663, December.
    5. 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.
    6. 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.
    7. Yufan Wang & Xingzhong Xu, 2023. "Homogeneity Test for Multiple Semicontinuous Data with the Density Ratio Model," Mathematics, MDPI, vol. 11(17), pages 1-28, September.
    8. 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.
    9. 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.
    10. Giovanni Compiani & Yuichi Kitamura, 2016. "Using mixtures in econometric models: a brief review and some new results," Econometrics Journal, Royal Economic Society, vol. 19(3), pages 95-127, October.
    11. Andr�s Farall & Ricardo Maronna & Tomás Tetzlaff, 2011. "A mixture model for the detection of Neosporosis without a gold standard," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(5), pages 913-926, February.
    12. Wang, Chunlin & Marriott, Paul & Li, Pengfei, 2017. "Testing homogeneity for multiple nonnegative distributions with excess zero observations," Computational Statistics & Data Analysis, Elsevier, vol. 114(C), pages 146-157.
    13. Jing Qin & Kung-Yee Liang, 2011. "Hypothesis Testing in a Mixture Case–Control Model," Biometrics, The International Biometric Society, vol. 67(1), pages 182-193, March.
    14. 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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