Electronic Journal of Statistics

Bayesian estimation in a high dimensional parameter framework

Denis Bosq and María D. Ruiz-Medina

Full-text: Open access

Abstract

Sufficient conditions are derived for the asymptotic efficiency and equivalence of componentwise Bayesian and classical estimators of the infinite-dimensional parameters characterizing $l^{2}$ valued Poisson process, and Hilbert valued Gaussian random variable models. Conjugate families are considered for the Poisson and Gaussian univariate likelihoods, in the Bayesian estimation of the components of such infinite-dimensional parameters. In the estimation of the functional mean of a Hilbert valued Gaussian random variable, sufficient and necessary conditions, that ensure a better performance of the Bayes estimator with respect to the classical one, are also obtained for the finite-sample size case. A simulation study is carried out to provide additional information on the relative efficiency of Bayes and classical estimators in a high-dimensional framework.

Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 1604-1640.

Dates
First available in Project Euclid: 8 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1410181226

Digital Object Identifier
doi:10.1214/14-EJS935

Mathematical Reviews number (MathSciNet)
MR3263132

Zentralblatt MATH identifier
1297.62188

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 65F15: Eigenvalues, eigenvectors

Keywords
Asymptotic relative efficiency Bayesian estimation Hilbert valued Gaussian random variable Hilbert valued Poisson process

Citation

Bosq, Denis; Ruiz-Medina, María D. Bayesian estimation in a high dimensional parameter framework. Electron. J. Statist. 8 (2014), no. 1, 1604--1640. doi:10.1214/14-EJS935. https://projecteuclid.org/euclid.ejs/1410181226


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References

  • [1] Antoniadis, A. and Sapatinas, T. (2003). Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes. Journal of Multivariate Analysis 87 133–158.
  • [2] Bali, J. L., Boente, G., Tyler, D. E. and Wang, J.-L. (2011). Robust functional principal components: A projection-pursuit approach. 39, 2795–3443.
  • [3] Basse, M., Diop, A. and Dabo-Niang, S. (2008). Mean square properties of a class of kernel density estimates for spatial functional random variables. Annales De L’I.S.U.P. Publications de l’Institut de Statistique de l’Université de Paris.
  • [4] Besse, P. C. and Cardot, H. (1996). Approximation spline de la prévision dún processus fonctionnel autorégressif dórdre 1. Canad. J. Statist. 24 467–487.
  • [5] Blanke, D. and Bosq, D. (2014). Bayesian prediction for stochastic processes. To appear in Sankhya.
  • [6] Bosq, D. (2000). Linear Processes in Function Spaces. Springer-Verlag, New York.
  • [7] Bosq, D. and Blanke, D. (2007). Inference and Predictions in Large Dimensions. John Wiley, Chichester, UK
  • [8] Da Prato, G. and Zabczyk, J. (2002). Second Order Partial Differential Equations in Hilbert spaces. Cambridge University Press, New York.
  • [9] Dautray, R. and Lions, J.-L. (1990). Mathematical Analysis and Numerical Methods for Science and Technology. Volume 3: Spectral Theory and Applications. Springer ISBN 978-3-540-66099-6.
  • [10] Fahrmeir, L. and Kneib, T. (2011). Bayesian Smoothing and Regression for Longitudinal, Spatial and Event History Data. Oxford University Press, New York.
  • [11] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis. Springer, New York.
  • [12] Khoshnevisan, D. (2007). Probability. American Mathematical Society, UE.
  • [13] Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation. Springer-Verlag, New York.
  • [14] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis (2nd edition). Springer-Verlag, New York.
  • [15] Ruiz-Medina, M. D. (2011). Spatial autoregressive and moving average Hilbertian processes. J. Multiv. Anal. 102 292–305.
  • [16] Ruiz-Medina, M. D. (2012). Spatial functional prediction from spatial autoregressive Hilbertian processes. Environmetrics 23 119–128.
  • [17] Ruiz-Medina, M. D. and Salmerón, R. (2010). Functional maximum-likelihood estimation of ARH(p) models. Stoch. Env. Res. Risk. Asses. 24 131–146.
  • [18] Shojaie, A. and Michailidis, G. (2009). Analysis of gene sets based on the underlying regulatory network. J. Comp. Biol. 16, 407–26.
  • [19] Shojaie, A. and Michailidis, G. (2010). Penalized likelihood methods for estimation of sparse high-dimensional directed acyclic graphs. Biometrika 97 519–538.
  • [20] Walsh, J. B. (1984). Regularity properties of a stochastic partial differential equation. In Seminar on Stochastic Processes (Gainestville, FL, 1983; Progress Prob. Statist. 7), Birkhäuser, Boston, MA, pp. 257–290.