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Bayesian modal regression with linear inequality constraints using mixture distributions

Author

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  • Jialei Yu

    (Beijing University of Technology, School of Statistics and Data Science, School of Mathematics, Statistics and Mechanics)

  • Min Xu

    (Beijing University of Technology, School of Statistics and Data Science, School of Mathematics, Statistics and Mechanics)

  • Jinyu Chen

    (Beijing University of Technology, School of Statistics and Data Science, School of Mathematics, Statistics and Mechanics)

Abstract

Modal regression focuses on the most likely values of the response variable given the regression variables, thereby uncovering important structures that may be overlooked by traditional regression approaches. In practical applications, prior knowledge can be incorporated into the model as constraints, which restrict the parameters space and improve the accuracy of parameter estimation. To this end, we propose Bayesian modal regression models with linear inequality constraints. Based on a flexible Gumbel distribution and a two-piece scale Student-t distribution, we develop two MCMC sampling algorithms for the parameter estimation. Simulation studies demonstrate that the proposed methods with linear inequality constraints yield lower root mean square errors in parameter estimation and produce narrower prediction intervals compared to their unconstrained counterparts. Furthermore, we demonstrate the effectiveness of proposed methods by applying to real-world murder rate data.

Suggested Citation

  • Jialei Yu & Min Xu & Jinyu Chen, 2026. "Bayesian modal regression with linear inequality constraints using mixture distributions," Computational Statistics, Springer, vol. 41(3), pages 1-26, April.
    • Handle: RePEc:spr:compst:v:41:y:2026:i:3:d:10.1007_s00180-026-01729-3
    DOI: 10.1007/s00180-026-01729-3
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    References listed on IDEAS

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    1. Qingyang Liu & Xianzheng Huang & Haiming Zhou, 2024. "The Flexible Gumbel Distribution: A New Model for Inference about the Mode," Stats, MDPI, vol. 7(1), pages 1-16, March.
    2. Randi Hjalmarsson & Lance Lochner, 2012. "The Impact of Education on Crime: International Evidence," ifo DICE Report, ifo Institute - Leibniz Institute for Economic Research at the University of Munich, vol. 10(2), pages 49-55, 08.
    3. Liu, Qingyang & Huang, Xianzheng & Bai, Ray, 2024. "Bayesian modal regression based on mixture distributions," Computational Statistics & Data Analysis, Elsevier, vol. 199(C).
    4. John F. Geweke, 1996. "Bayesian Inference for Linear Models Subject to Linear Inequality Constraints," Springer Books, in: Jack C. Lee & Wesley O. Johnson & Arnold Zellner (ed.), Modelling and Prediction Honoring Seymour Geisser, pages 248-263, Springer.
    5. repec:ces:ifodic:v:10:y:2012:i:2:p:18946535 is not listed on IDEAS
    6. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1683, August.
    7. Jochen Einbeck & Gerhard Tutz, 2006. "Modelling beyond regression functions: an application of multimodal regression to speed–flow data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 55(4), pages 461-475, August.
    8. repec:bla:jfinan:v:58:y:2003:i:4:p:1651-1684 is not listed on IDEAS
    9. Randi Hjalmarsson & Lance Lochner, 2012. "The Impact of Education on Crime: International Evidence," ifo DICE Report, ifo Institute - Leibniz Institute for Economic Research at the University of Munich, vol. 10(02), pages 49-55, August.
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