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Bayesian ridge estimators based on copula-based joint prior distributions for regression coefficients

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Abstract

Ridge regression is a widely used method to mitigate the multicollinearly problem often arising in multiple linear regression. It is well known that the ridge regression estimator can be derived from the Bayesian framework by the posterior mode under a multivariate normal prior. However, the ridge regression model with a copula-based multivariate prior model has not been employed in the Bayesian framework. Motivated by the multicollinearly problem due to an interaction term, we adopt a vine copula to construct the copula-based joint prior distribution. For selected copulas and hyperparameters, we propose Bayesian ridge estimators and credible intervals for regression coefficients. A simulation study is carried out to compare the performance of four different priors (the Clayton, Gumbel, and Gaussian copula priors, and the tri-variate normal prior) on the regression coefficients. Our simulation studies demonstrate that the Archimedean (Clayton and Gumbel) copula priors give more accurate estimates in the presence of multicollinearity compared with the other priors. Finally, a real dataset is analyzed, where the Bayesian ridge estimators and some frequentist estimators are compared.

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Acknowledgements

The authors thank Editor, Associate Editor, and two referees for their valuable suggestions that improved the paper. This research was supported by JSPS KAKENHI Grant Number JP21K12127.

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Authors and Affiliations

  1. Department of Clinical Medicine (Biostatistics), School of Pharmacy, Kitasato University, Tokyo, 108-8641, Japan

    Hirofumi Michimae

  2. Biostatistics Center, Kurume University, 67 Asahi-machi, Kurume, Japan

    Takeshi Emura

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  1. Hirofumi Michimae
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  2. Takeshi Emura
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Correspondence to Hirofumi Michimae.

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Michimae, H., Emura, T. Bayesian ridge estimators based on copula-based joint prior distributions for regression coefficients. Comput Stat 37, 2741–2769 (2022). https://doi.org/10.1007/s00180-022-01213-8

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