## Abstract

Let X|μ ∼ N_{p}(μ, v_{x}I) and Y|μ∼N _{p}(μ, v_{y}I) be independent p-dimensional multivariate normal vectors with common unknown mean μ. Based on only observing X = x, we consider the problem of obtaining a predictive density p̂(y|x) for Y that is close to p(y|μ) as measured by expected Kullback-Leibler loss. A natural procedure for this problem is the (formal) Bayes predictive density p̂U(y|x) under the uniform prior πU(μ) ≡ 1, which is best invariant and minimax. We show that any Bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is super-harmonic or whose square root is superharmonic. This yields wide classes of minimax procedures that dominate p̂U(y|x), including Bayes predictive densities under superharmonic priors. Fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described.

Original language | English (US) |
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Pages (from-to) | 78-91 |

Number of pages | 14 |

Journal | Annals of Statistics |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2006 |

Externally published | Yes |

## Keywords

- Bayes rules
- Heat equation
- Inadmissibility
- Multiple shrinkage
- Multivariate normal
- Prior distributions
- Shrinkage estimation
- Superharmonic marginals
- Superharmonic priors
- Unbiased estimate of risk

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty