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For two vast families of mixture distributions and a given prior, we provide unified representations of posterior and predictive distributions. Model applications presented include bivariate mixtures of Gamma distributions labeled as Kibble-type, non-central Chi-square and F distributions, the distribution of R2 in multiple regression, variance mixture of normal distributions, and mixtures of location-scale exponential distributions including the multivariate Lomax distribution. An emphasis is also placed on analytical representations and the relationships with a host of existing distributions and several hypergeometric functions of one or two variables. [ABSTRACT FROM AUTHOR]

We describe a hierarchical Bayesian approach for inference about a parameter θ lower-bounded by α with uncertain α, derive some basic identities for posterior analysis about (θ , α) , and provide illustrations for normal and Poisson models. For the normal case with unknown mean θ and known variance σ 2 , we obtain Bayes estimators of θ that take values on R , but that are equally adapted to a lower-bound constraint in being minimax under squared error loss for the constrained problem. [ABSTRACT FROM AUTHOR]

We describe a hierarchical Bayesian approach for inference about a parameter θ lower-bounded by α with uncertain α, derive some basic identities for posterior analysis about (θ , α) , and provide illustrations for normal and Poisson models. For the normal case with unknown mean θ and known variance σ 2 , we obtain Bayes estimators of θ that take values on R , but that are equally adapted to a lower-bound constraint in being minimax under squared error loss for the constrained problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with prediction for Gamma models, and more specifically the estimation of a predictive density for Y ∼ G a ( α 2 , β ) under Kullback–Leibler loss, based on X ∼ G a ( α 1 , β ) . The main focus pertains to situations where there is a parametric constraint of the form β ∈ C = ( a , b ) . We obtain representations for Bayes predictive densities and the minimum risk equivariant predictive density in the unconstrained problem. It is shown that the generalized Bayes estimator against the truncation of the non-informative prior onto C dominates the minimum risk equivariant predictive density and is minimax whenever a = 0 or b = ∞ . Analytical comparisons of plug-in predictive densities Ga ( α 2 , β ˆ ) , which include the predictive mle density, are obtained with results applying as well for point estimation under dual entropy loss β β ˆ − log ( β β ˆ ) − 1 . Numerical evaluations confirm that such predictive densities are much less efficient than some Bayesian alternatives in exploiting the parametric restriction. Finally, it is shown that variance expansion improvements of the form G a ( α 2 k , k β ˆ ) of plug-in predictive densities can always be found for a subset of k > 1 values and non-degenerate β ˆ . [ABSTRACT FROM AUTHOR]

Our investigation concerns the estimation of predictive densities and a study of efficiency as measured by the frequentist risk of such predictive densities with integrated squared error loss. Our findings relate to a d -variate spherically symmetric observable X ∼ p X ( ‖ x − μ ‖ 2 ) and the objective of estimating the density of Y ∼ q Y ( ‖ y − μ ‖ 2 ) based on X . We describe Bayes estimation, minimum risk equivariant estimation (MRE), and minimax estimation. We focus on the risk performance of the benchmark minimum risk equivariant estimator, plug-in estimators, and plug-in type estimators with expanded scale. For the multivariate normal case, we make use of a duality result with a point estimation problem bringing into play reflected normal loss. In three or more dimensions (i.e., d ≥ 3 ), we show that the MRE predictive density estimator is inadmissible and provide dominating estimators. This brings into play Stein-type results for estimating a multivariate normal mean with a loss which is a concave and increasing function of ‖ μ ˆ − μ ‖ 2 . We also study the phenomenon of improvement on the plug-in density estimator of the form q Y ( ‖ y − a X ‖ 2 ) , 0 < a ≤ 1 , by a subclass of scale expansions 1 c d q Y ( ‖ ( y − a X ) / c ‖ 2 ) with c > 1 , showing in some cases, inevitably for large enough d , that all choices c > 1 are dominating estimators. Extensions are obtained for scale mixture of normals including a general inadmissibility result of the MRE estimator for d ≥ 3 . [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with estimation of a predictive density with parametric constraints under Kullback–Leibler loss. When an invariance structure is embedded in the problem, general and unified conditions for the minimaxity of the best equivariant predictive density estimator are derived. These conditions are applied to check minimaxity in various restricted parameter spaces in location and/or scale families. Further, it is shown that the generalized Bayes estimator against the uniform prior over the restricted space is minimax and dominates the best equivariant estimator in a location family when the parameter is restricted to an interval of the form . Similar findings are obtained for scale parameter families. Finally, the presentation is accompanied by various observations and illustrations, such as normal, exponential location, and gamma model examples. [Copyright &y& Elsevier]

Abstract: For independently distributed observables: , we consider estimating the vector with loss under the constraint , with known . In comparing the risk performance of Bayesian estimators associated with uniform priors on spheres of radius centered at with that of the maximum likelihood estimator , we make use of Stein’s unbiased estimate of risk technique, Karlin’s sign change arguments, and a conditional risk analysis to obtain for a fixed necessary and sufficient conditions on for to dominate . Large sample determinations of these conditions are provided. Both cases where all such ’s and cases where no such ’s dominate are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator dominates if and only if with , improving on the previously known sufficient condition of Marchand and Perron (2001)  for which . Finally, we improve upon a universal dominance condition due to Marchand and Perron, by establishing that all Bayesian estimators with spherically symmetric and supported on the parameter space dominate whenever with . [Copyright &y& Elsevier]

For the problem of estimating under squared error loss the location parameter of a p-variate spherically symmetric distribution where the location parameter lies in a ball of radius m, a general sufficient condition for an estimator to dominate the maximum likelihood estimator is obtained. Dominance results are then made explicit for the case of a multivariate student distribution with d degrees of freedom and, in particular, we show that the Bayes estimator with respect to a uniform prior on the boundary of the parameter space dominates the maximum likelihood estimator whenever m⩽√ of p and d⩾p. The sufficient condition m⩽√ of p matches the one obtained by Marchand and Perron (Ann. Statist. 29 (2001) 1078) in the normal case with identity covariance matrix. Furthermore, we derive an explicit class of estimators which, for m<√ of p, dominate the maximum likelihood estimator simultaneously for the normal distribution with identity covariance matrix and for all multivariate student distributions with d degrees of freedom, d⩾p. Finally, we obtain estimators which dominate the maximum likelihood estimator simultaneously for all distributions in the subclass of scale mixtures of normals for which the scaling random variable is bounded below by some positive constant with probability one. [Copyright &y& Elsevier]