For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the 100(1 − α)% Bayesian HPD credible set associated with priors which are truncations of flat priors onto the restricted parameter space. Various new properties are obtained. Namely, we identify precisely where the minimum coverage is obtained and we show that this minimum coverage is bounded between 1 − 3α 2 and 1 − 3α 2 + α 2 1+α ; with the lower bound 1 − 3α 2 improving (for α ≤ 1/3) on the previously established ([9]; [8]) lower bound 1−α 1+α . Several illustrative examples are given.AMS 2000 subject classifications: 62F10, 62F30, 62C10, 62C15, 35Q15, 45B05, 42A99.
Based on X ∼ N d (θ, σ 2 X I d ), we study the efficiency of predictive densities under α−divergence loss L α for estimating the density ofWe identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension d, the variances σ 2 X and σ 2 Y , the choice of loss L α ; α ∈ (−1, 1). The findings also apply to a large number of plug-in densities, as well as for restricted parameter spaces with θ ∈ Θ ⊂ R d . The theoretical findings are accompanied by various observations, illustrations, and implications dealing for instance with robustness with respect to the model variances and simultaneous dominance with respect to the loss.
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