## Abstract

We complement the results of Fourdrinier, Mezoued and Strawderman in [5] who considered Bayesian estimation of the location parameter θ of a random vector X having a unimodal spherically symmetric density f({norm of matrix}x-θ {norm of matrix}^{2}) for a spherically symmetric prior density π({norm of matrix}θ {norm of matrix}^{2}). In [5], expressing the Bayes estimator as δ_{π}(X)=X+∇M({norm of matrix}X {norm of matrix}^{2})/m({norm of matrix}X {norm of matrix}^{2}), where m is the marginal associated to f({norm of matrix}x-θ {norm of matrix}^{2}) and M is the marginal with respect to, it was shown that, under quadratic loss, if the sampling density f({norm of matrix}x-θ {norm of matrix}^{2}) belongs to the Berger class (i.e. there exists a positive constant c such that F(t)/f(t)≥c for all t), conditions, dependent on the monotonicity of the ratio F(t)/f(t), can be found on π in order that δ_{π}(X) is minimax. The main feature of this paper is that, in the case where F(t)/f(t) is nonincreasing, if π({norm of matrix}θ {norm of matrix}^{2}) is a superharmonic power prior of the form {norm of matrix}θ {norm of matrix}^{-2k} with k>0, the membership of the sampling density to the Berger class can be droped out. Also, our techniques are different from those in [5]. First, writing δ_{π}(X)=X+g(X) with g(X)∝∇M({norm of matrix}X {norm of matrix}^{2}) /m({norm of matrix}X {norm of matrix}^{2}), we follow Brandwein and Strawderman [4] proving that, for some b>0, the function h=bΔM/m is subharmonic and satisfies {norm of matrix}g {norm of matrix}^{2}/2 ≤ -h≤ -divg. Also, we adapt their approach using the fact that is nonincreasing in R for any θ∈ℝ^{p}, when V_{θ,R} is the uniform distribution on the ball B_{θ,R} of radius R and centered at θ. Examples illustrate the theory.

Original language | English (US) |
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Pages (from-to) | 717-741 |

Number of pages | 25 |

Journal | Electronic Journal of Statistics |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Bayes estimators
- Location parameter
- Minimax estimators
- Power priors
- Quadratic loss
- Spherically symmetric distributions
- Unimodal densities