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On predictive density estimation for location families under integrated squared error loss. (English) Zbl 1327.62054

Summary: 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 \(X \sim p_X(\| x - \mu \|^2)\) and the objective of estimating the density of \(Y \sim q_Y(\| y - \mu \|^2)\) 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., \(d \geq 3\)), 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 \(\| \hat{\mu} - \mu \|^2\). We also study the phenomenon of improvement on the plug-in density estimator of the form \(q_Y(\| y - a X \|^2), 0 < a \leq 1\), by a subclass of scale expansions \(\frac{1}{c^d} q_Y(\|(y - a X) / c \|^2)\) 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 \(d \geq 3\).

MSC:

62C20 Minimax procedures in statistical decision theory
62C86 Statistical decision theory and fuzziness
62F10 Point estimation
62F15 Bayesian inference
62F30 Parametric inference under constraints
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References:

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