### Abstract

We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function. Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from that of training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant. Then, we consider a class of priors for the mean parameters depending on the future covariance matrix. With such a prior, we can construct a Bayesian predictive distribution dominating that based on the uniform prior. Lastly, applying this result to the prediction of response variables in the Normal linear regression model, we show that there exists a Bayesian predictive distribution dominating that based on the uniform prior. Minimaxity of these Bayesian predictions follows from these results.

Original language | English |
---|---|

Pages (from-to) | 1888-1905 |

Number of pages | 18 |

Journal | Journal of Multivariate Analysis |

Volume | 99 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2008 Oct |

Externally published | Yes |

### Fingerprint

### Keywords

- 62C10
- 62F07
- 62F15
- 62J07
- Bayesian prediction
- Kullback-Leibler divergence
- Minimaxity
- Normal regression
- primary
- secondary
- Shrinkage estimation
- Superharmonic function

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Numerical Analysis
- Statistics and Probability

### Cite this

*Journal of Multivariate Analysis*,

*99*(9), 1888-1905. https://doi.org/10.1016/j.jmva.2008.01.014

**Bayesian shrinkage prediction for the regression problem.** / Kobayashi, Kei; Komaki, Fumiyasu.

Research output: Contribution to journal › Article

*Journal of Multivariate Analysis*, vol. 99, no. 9, pp. 1888-1905. https://doi.org/10.1016/j.jmva.2008.01.014

}

TY - JOUR

T1 - Bayesian shrinkage prediction for the regression problem

AU - Kobayashi, Kei

AU - Komaki, Fumiyasu

PY - 2008/10

Y1 - 2008/10

N2 - We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function. Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from that of training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant. Then, we consider a class of priors for the mean parameters depending on the future covariance matrix. With such a prior, we can construct a Bayesian predictive distribution dominating that based on the uniform prior. Lastly, applying this result to the prediction of response variables in the Normal linear regression model, we show that there exists a Bayesian predictive distribution dominating that based on the uniform prior. Minimaxity of these Bayesian predictions follows from these results.

AB - We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function. Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from that of training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant. Then, we consider a class of priors for the mean parameters depending on the future covariance matrix. With such a prior, we can construct a Bayesian predictive distribution dominating that based on the uniform prior. Lastly, applying this result to the prediction of response variables in the Normal linear regression model, we show that there exists a Bayesian predictive distribution dominating that based on the uniform prior. Minimaxity of these Bayesian predictions follows from these results.

KW - 62C10

KW - 62F07

KW - 62F15

KW - 62J07

KW - Bayesian prediction

KW - Kullback-Leibler divergence

KW - Minimaxity

KW - Normal regression

KW - primary

KW - secondary

KW - Shrinkage estimation

KW - Superharmonic function

UR - http://www.scopus.com/inward/record.url?scp=52749092488&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52749092488&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2008.01.014

DO - 10.1016/j.jmva.2008.01.014

M3 - Article

AN - SCOPUS:52749092488

VL - 99

SP - 1888

EP - 1905

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 9

ER -