Abstract
Rating data are a kind of ordinal categorical data routinely collected in survey sampling. The response value in such applications is confined to a finite number of ordered categories. Due to population heterogeneity, the respondents may have several different rating styles. A finite mixture model is thus most suitable to fit datasets of this nature. In this paper, we propose a two-component mixture of shifted binomial distributions for rating data. We show that this model is identifiable and propose a numerically stable penalized likelihood approach for parameter estimation. We adapt an expectation-maximization algorithm for the penalized maximum likelihood estimation. Our simulation results show that the penalized maximum likelihood estimator is consistent and effective. We fit the proposed model and other models in the literature to some real-world datasets and find the proposed model can have much better fits.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11701071, 11871419), the Natural Science and Engineering Research Council (Grant No. 2019–04204) and the Scientific Research Projects of Dongbei University of Finance and Economics (Grand No. 20210261). We thank the AE and referee for helpful comments and suggestions.
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Li, S., Chen, J. Mixture of shifted binomial distributions for rating data. Ann Inst Stat Math 75, 833–853 (2023). https://doi.org/10.1007/s10463-023-00865-7
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DOI: https://doi.org/10.1007/s10463-023-00865-7