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On improved predictive density estimation with parametric constraints. (English) Zbl 1274.62079

Summary: We consider the problem of predictive density estimation for normal models under Kullback-Leibler loss (KL loss) when the parameter space is constrained to a convex set. More particularly, we assume that \(X\sim \mathcal N_p(\mu, v_xI)\) is observed and that we wish to estimate the density of \(Y\sim \mathcal N_p(\mu, v_yI)\) under KL loss when \(\mu \) is restricted to the convex set \(C\subset \mathbb R^{p}\). We show that the best unrestricted invariant predictive density estimator \(\hat p_{U}\) is dominated by the Bayes estimator \(\hat p_{\pi_{C}}\) associated to the uniform prior \(\pi _{C}\) on \(C\). We also study so called plug-in estimators, giving conditions under which domination of one estimator of the mean vector \(\mu \) over another under the usual quadratic loss, translates into a domination result for certain corresponding plug-in density estimators under KL loss. Risk comparisons and domination results are also made for comparisons of plug-in estimators and Bayes predictive density estimators. Additionally, minimaxity and domination results are given for the cases where: (i) \(C\) is a cone, and (ii) \(C\) is a ball.

MSC:

62C15 Admissibility in statistical decision theory
62C20 Minimax procedures in statistical decision theory
62F10 Point estimation
62H12 Estimation in multivariate analysis
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References:

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