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

Abstract

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.

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

“…Note that, due to the superharmonicity property, our priors cannot be proper (see [8]). However, neither our priors nor our densities need to be mixture of normals.…”

Bayes minimax estimation of the multivariate normal mean vector for the case of common unknown variance

“…Following Strawderman [9] and Berger [1], Maruyama [5] found an extended class of generalized Bayes minimax admissible estimators for the mean vector of a scale mixture of multivariate normal distributions. Fourdrinier et al [3] extended Maruyama's result [5] and obtained a large class of Bayes minimax estimators.…”

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

“…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.…”

“…[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).…”

“…[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].…”