Electronic Journal of Statistics

On improved predictive density estimation with parametric constraints

Dominique Fourdrinier, Éric Marchand, Ali Righi, and William E. Strawderman

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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 $X\sim {\cal N}_{p}(\mu,v_{x}I)$ is observed and that we wish to estimate the density of $Y\sim {\cal N}_{p}(\mu,v_{y}I)$ under KL loss when μ is restricted to the convex set Cp. We show that the best unrestricted invariant predictive density estimator U is dominated by the Bayes estimator πC associated to the uniform prior πC on C. We also study so called plug-in estimators, giving conditions under which domination of one estimator of the mean vector μ 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.

Article information

Electron. J. Statist., Volume 5 (2011), 172-191.

First available in Project Euclid: 14 April 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C15: Admissibility 62C20: Minimax procedures 62F10: Point estimation 62H12: Estimation

Predictive estimation risk function quadratic loss Kullback-Leibler loss uniform priors Bayes estimators convex sets cones multivariate normal


Fourdrinier, Dominique; Marchand, Éric; Righi, Ali; Strawderman, William E. On improved predictive density estimation with parametric constraints. Electron. J. Statist. 5 (2011), 172--191. doi:10.1214/11-EJS603. https://projecteuclid.org/euclid.ejs/1302784852

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