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Enriched standard conjugate priors and the right invariant prior for Wishart distributions

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  • Oda, Hidemasa
  • Komaki, Fumiyasu

Abstract

The prediction of the variance–covariance matrix of the multivariate normal distribution is important in the multivariate analysis. We investigated Bayesian predictive distributions for Wishart distributions under the Kullback–Leibler divergence. The conditional reducibility of the family of Wishart distributions enables us to decompose the risk of a Bayesian predictive distribution. We considered a recently introduced class of prior distributions, which is called the family of enriched standard conjugate prior distributions, and compared the Bayesian predictive distributions based on these prior distributions. Furthermore, we studied the performance of the Bayesian predictive distribution based on the reference prior distribution in the family and showed that there exists a prior distribution in the family that dominates the reference prior distribution. Our study provides new insight into the multivariate analysis when there exists an ordered inferential importance for the independent variables.

Suggested Citation

  • Oda, Hidemasa & Komaki, Fumiyasu, 2023. "Enriched standard conjugate priors and the right invariant prior for Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    • Handle: RePEc:eee:jmvana:v:193:y:2023:i:c:s0047259x22000963
    DOI: 10.1016/j.jmva.2022.105105
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    References listed on IDEAS

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    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Komaki, Fumiyasu, 2009. "Bayesian predictive densities based on superharmonic priors for the 2-dimensional Wishart model," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2137-2154, November.
    3. repec:dau:papers:123456789/1908 is not listed on IDEAS
    4. Guido Consonni & Piero Veronese, 2001. "Conditionally Reducible Natural Exponential Families and Enriched Conjugate Priors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(2), pages 377-406, June.
    5. Chih-Wen Hsu & Marick Sinay & John Hsu, 2012. "Bayesian estimation of a covariance matrix with flexible prior specification," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(2), pages 319-342, April.
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