Electronic Journal of Statistics

Asymptotically minimax prediction in infinite sequence models

Keisuke Yano and Fumiyasu Komaki

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We study asymptotically minimax predictive distributions in infinite sequence models. First, we discuss the connection between prediction in an infinite sequence model and prediction in a function model. Second, we construct an asymptotically minimax predictive distribution for the setting in which the parameter space is a known ellipsoid. We show that the Bayesian predictive distribution based on the Gaussian prior distribution is asymptotically minimax in the ellipsoid. Third, we construct an asymptotically minimax predictive distribution for any Sobolev ellipsoid. We show that the Bayesian predictive distribution based on the product of Stein’s priors is asymptotically minimax for any Sobolev ellipsoid. Finally, we present an efficient sampling method from the proposed Bayesian predictive distribution.

Article information

Electron. J. Statist. Volume 11, Number 2 (2017), 3165-3195.

Received: June 2016
First available in Project Euclid: 11 September 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62G20: Asymptotic properties
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

Adaptivity Kullback–Leibler divergence nonparametric statistics predictive distribution Stein’s prior

Creative Commons Attribution 4.0 International License.


Yano, Keisuke; Komaki, Fumiyasu. Asymptotically minimax prediction in infinite sequence models. Electron. J. Statist. 11 (2017), no. 2, 3165--3195. doi:10.1214/17-EJS1312. https://projecteuclid.org/euclid.ejs/1505116877

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  • [1] Belitser, E. andGhosal, S. (2003). Adaptive Bayesian inference on the mean of an infinite-dimensional normal, distribution.Ann. Statist.31536–559.
  • [2] Brown, L. D., George, E. I. andXu, X. (2008). Admissible predictive density, estimation.Ann. Statist.361156–1170.
  • [3] Brown, L. D. andZhao, L. H. (2009). Estimators for Gaussian models having a block-wise, structure.Statist. Sinica19885–903.
  • [4] Cai, T., Low, M. andZhao, L. (2000). Sharp adaptive estimation by a blockwise method Technical Report, Warton School, University of Pennsylvania, Philadelphia.
  • [5] Cavalier, L. andTsybakov, A. B. (2001). Penalized blockwise Stein’s method, monotone oracles and sharp adaptive, estimation.Math. Methods Statist.10247–282.
  • [6] Dudley, R. M., (2002).Real analysis and probability. Cambridge University Press, Cambridge.
  • [7] Efromovich, S. andPinsker, M. (1984). Learning algorithm for nonparmetric, filtering.Automation and Remote Control111434–1440.
  • [8] George, E. I., Liang, F. andXu, X. (2006). Improved minimax predictive densities under Kullback-Leibler, loss.Ann. Statist.3478–91.
  • [9] Komaki, F. (2001). A shrinkage predictive distribution for multivariate normal, observables.Biometrika88859–864.
  • [10] Kubokawa, T. (1991). An approach to improving the James-Stein, estimator.J. Multivariate Anal.36121–126.
  • [11] Mandelbaum, A. (1984). All admissible linear estimators of the mean of a Gaussian distribution on a Hilbert, space.Ann. Statist.121448–1466.
  • [12] Mukherjee, G. andJohnstone, I. M. (2015). Exact minimax estimation of the predictive density in sparse Gaussian, models.Ann. Statist.43937–961.
  • [13] Philippe, A. (1997). Simulation of right and left truncated gamma distributions by, mixtures.Stat. Comput.7173–181.
  • [14] Pinsker, M. S. (1980). Optimal filtration of square-integrable signals in Gaussian, noise.16120–133.
  • [15] Pollard, D., (2002).A User’s Guide to Measure Theoretic Probability. Cambridge University Press, Cambridge.
  • [16] Tsybakov, A., (2009).Introduction to Nonparametric Estimation. Springer, New York.
  • [17] Wasserman, L., (2007).All of Nonparametric Statistics, 3rd ed. Springer, New York.
  • [18] Williams, D., (1991).Probability with Martingale. Cambridge University Press, Cambridge.
  • [19] Xu, X. andLiang, F. (2010). Asymptotic minimax risk of predictive density estimation for non-parametric, regression.Bernoulli16543–560.
  • [20] Xu, X. andZhou, D. (2011). Empirical Bayes predictive densities for high-dimensional normal, models.J. Multivariate Anal.1021417–1428.
  • [21] Zhao, L. H. (2000). Bayesian aspects of some nonparametric, problems.Ann. Statist.28532–552.