Bayes minimax estimation under power priors of location parameters for a wide class of spherically symmetric distributions

Dominique Fourdrinier, Fatiha Mezoued, William E. Strawderman

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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 δπ(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 δπ(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 δπ(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∈ℝp, when Vθ,R is the uniform distribution on the ball Bθ,R of radius R and centered at θ. Examples illustrate the theory.

Original languageEnglish (US)
Pages (from-to)717-741
Number of pages25
JournalElectronic Journal of Statistics
Volume7
Issue number1
DOIs
StatePublished - 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

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