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June 2015 Exact minimax estimation of the predictive density in sparse Gaussian models
Gourab Mukherjee, Iain M. Johnstone
Ann. Statist. 43(3): 937-961 (June 2015). DOI: 10.1214/14-AOS1251


We consider estimating the predictive density under Kullback–Leibler loss in an $\ell_{0}$ sparse Gaussian sequence model. Explicit expressions of the first order minimax risk along with its exact constant, asymptotically least favorable priors and optimal predictive density estimates are derived. Compared to the sparse recovery results involving point estimation of the normal mean, new decision theoretic phenomena are seen. Suboptimal performance of the class of plug-in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk. We find that minimax optimal strategies lie outside the Gaussian family but can be constructed with threshold predictive density estimates. Novel minimax techniques involving simultaneous calibration of the sparsity adjustment and the risk diversification mechanisms are used to design optimal predictive density estimates.


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Gourab Mukherjee. Iain M. Johnstone. "Exact minimax estimation of the predictive density in sparse Gaussian models." Ann. Statist. 43 (3) 937 - 961, June 2015.


Received: 1 January 2014; Revised: 1 June 2014; Published: June 2015
First available in Project Euclid: 15 May 2015

zbMATH: 1328.62058
MathSciNet: MR3346693
Digital Object Identifier: 10.1214/14-AOS1251

Primary: 62C20
Secondary: 60G25 , 62M20 , 91G70

Keywords: high-dimensional , minimax , mutual information , plug-in risk , predictive density , risk diversification , Sparsity , thresholding

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 3 • June 2015
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