Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family

Let X ∼ f ( x − 2 ) and let (X) be the generalized Bayes estimator of with respect to a spherically symmetric prior, ( 2 ), for loss − 2 . We show that if (t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator 0 (X)=X under certain conditions on f ( ). The class of priors includes priors of the form 1 A+ 2 k for k p 2 − 1 and hence includes the fundamental harmonic prior 1 p−2 . The class of sampling distributions includes certain variance mixtures of normals and other functions f (t) of the form e − t and e − t+ (t ) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as (X) = 1 − ar(t) t X.

Section: Discussionsupporting

confidence: 79%

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Fourdrinier¹,

Strawderman²

2008

“…Comment (Admissibility): For priors with mixing distribution h satisfying (3.5) and (3.7) an argument as in Maruyama [9] using Brown [5] and a Tauberian theorem suggests that the resulting generalized Bayes estimator is admissible if 0. A referee called to our attention that, recently, Maruyama and Takemura [10] have verified this under additional conditions which imply, in our setting, that E [ X 3 ] < ∞.…”

Section: Lemma 32 Assume That the Mixing Density G Of The Sampling mentioning

confidence: 65%

Bayes minimax estimators of the mean of a scale mixture of multivariate normal distributions

Fourdrinier

Kortbi

Strawderman

2008

Bayes estimation of the mean of a variance mixture of multivariate normal distributions is considered under sum of squared errors loss. We find broad class of priors (also in the variance mixture of normal class) which result in proper and generalized Bayes minimax estimators. This paper extends the results of Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264] in a manner similar to that of Maruyama [Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distribution, J. Multivariate Anal. 21 (2003) 69-78] but somewhat more in the spirit of Fourdrinier et al. [On the construction of bayes minimax estimators, Ann. Statist. 26 (1998) 660-671] for the normal case, in the sense that we construct classes of priors giving rise to minimaxity. A feature of this paper is that in certain cases we are able to construct proper Bayes minimax estimators satisfying the properties and bounds in Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264].We also give some insight into why Strawderman's results do or do not seem to apply in certain cases. In cases where it does not apply, we give minimax estimators based on Berger's [Minimax estimation of location vectors for a wide class of densities, Ann. Statist. 3 (1975) 1318-1328] results. A main condition for minimaxity is that the mixing distributions of the sampling distribution and the prior distribution satisfy a monotone likelihood ratio property with respect to a scale parameter.

“…Recently, [6,10,13,18] gave conditions for minimaxity of generalized Bayes estimators of the location vector of a spherically symmetric distribution under squared error loss. [13,18] consider the general spherical case whereas [10] consider the scale mixture of normals.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[13,18] consider the general spherical case whereas [10] consider the scale mixture of normals. The results in [10,13,18] do not cover the estimation problem considered here since the model sufficient statistics in (1.1) are (X, S), hence the corresponding posterior distribution will also depend on (X, S). The spherically symmetric distribution cases discussed in [10,13,16,18] do not have a scale parameter nor do they contain a "residual vector" (the spherically symmetric distribution analog of S) as in the case of spherically (or more generally elliptically) symmetric distributions discussed in [12].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generalized Bayes minimax estimators of the mean of multivariate normal distribution with unknown variance

Wells

Zhou

2008