Skip to main content
Log in

On Predictive Density Estimation under α-Divergence Loss

  • Published:
Mathematical Methods of Statistics Aims and scope Submit manuscript

Abstract

Based on X ∼ Nd(θ, σ 2 X Id), we study the efficiency of predictive densities under α-divergence loss Lα for estimating the density of Y ∼ Nd(θ, σ 2 Y Id). We identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension d, the variances σ 2 X and σ 2 Y , the choice of loss Lα; α ∈ (−1, 1). The findings also apply to a large number of plug-in densities, as well as for restricted parameter spaces with θ ∈ Θ ⊂ ℝd. The theoretical findings are accompanied by various observations, illustrations, and implications dealing for instance with robustness with respect to the model variances and simultaneous dominance with respect to the loss.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Aitchison, “Goodness of Prediction Fit”, Biometrika 62, 547–554 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Aitchison and I. R. Dunsmore, Statistical Prediction Analysis (Cambridge Univ. Press, Cambridge, 1975).

    Book  MATH  Google Scholar 

  3. A. J. Baranchik, “A family of Minimax Estimators of the Mean of a Multivariate Normal Distribution”, Ann. Math. Statist. 41, 642–645 (1970).

    Article  MathSciNet  Google Scholar 

  4. J. O. Berger, “Minimax Estimation of a Multivariate Normal Mean under Polynomial Loss”, J. Multivariate Anal. 8, 173–180 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  5. L. D. Brown, E. I. George, and X. Xu, “Admissible Predictive Density Estimation”, Ann. Statist. 36, 1156–1170 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. M. Corcuera and F. Giummolè, “A Generalized Bayes Rule for Prediction”, Scand. J. Statist. 26, 265–279 (1999A).

    Article  MathSciNet  MATH  Google Scholar 

  7. J. M. Corcuera and F. Giummolè, “On the Relationship between α Connections and the Asymptotic Properties of Predictive Distributions”, Bernoulli 5, 163–176 (1999B).

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Csiszàr, “Information-Type Measures of Difference of Probability Distributions and Indirect Observations”, Studia Sci. Math. Hungar. 2, 299–318 (1967).

    MathSciNet  MATH  Google Scholar 

  9. D. Fourdrinier, É. Marchand, A. Righi, and W. E. Strawderman, “On Improved Predictive Density Estimation with Parametric Constraints”, Electron. J. Statist. 5, 172–191 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Fourdrinier, I. Ouassou, and W. E. Strawderman, “Estimation of a Mean Vector Under Quartic Loss”, J. Statist. Plann, Inference 138, 3841–3857 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. E. I. George, F. Liang, and X. Xu, “Improved Minimax Predictive Densities under Kullback-Leibler Loss”, Ann. Statist. 34, 78–91 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Ghosh, V. Mergel, and G. S. Datta, “Estimation, Prediction and the Stein Phenomenon under Divergence Loss”, J. Multivariate Anal. 99, 1941=–1961 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  13. T. Kubokawa, É. Marchand, and W. E. Strawderman, “On Predictive Density Estimation for Location Families under Integrated Absolute Value Loss”, Bernoulli 23, 3197–3212 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  14. T. Kubokawa, É. Marchand, and W. E. Strawderman, “On Predictive Density Estimation for Location Families under Integrated Squared Error Loss”, J. Multivariate Anal. 142, 57–74 (2015A)

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Kubokawa, É. Marchand, and W. E. Strawderman, “On Improved Shrinkage Estimators under Concave Loss”, Statist. Probab. Lett. 96, 241–246 (2015B).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. L’Moudden, É. Marchand, O. Kortbi, and W. E. Strawderman, “On Predictive Density Estimation for Gamma Models with Parametric Constraints”, J. Statist. Plann. Inference 185, 56–68 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  17. É. Marchand, F. Perron, and I. Yadegari, “On Estimating a Bounded Normal Mean with Applications to Predictive Density Estimation”, Electron. J. Statist. 11, 2002–2025 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. É. Marchand and N. Sadeghkhani, “On Predictive Density Estimation with Additional Information”, Electron. J. Statist. (in press) (2017).

  19. É. Marchand and W. E. Strawderman, “A Unified Minimax Result for Restricted Parameter Spaces”, Bernoulli 18, 635–643 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Maruyama and T. Ohnishi, “Harmonic Bayesian Prediction under α-Divergence”, arXiv:1605.05899v4 (2017).

  21. Y. Maruyama and W. E. Strawderman, “Bayesian Predictive Densities for Linear Regression Models under α-Divergence Loss: Some Results and Open Problems”, in Contemporary Developments in Bayesian analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman, IMS Collections (2012), Vol. 8, pp. 42–56.

    MathSciNet  MATH  Google Scholar 

  22. T. Yanagimoto and T. Ohnishi, “Bayesian Prediction of a Density Function in Terms of e-Mixture”, J. Statist. Plann. Inference 139, 3064–3075 (2009).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Eric Marchand’s research is supported in part by the Natural Sciences and Engineering Research Council of Canada. We thank Bill Strawderman who provided the lower bound in (2.19), as well as Jean Vaillancourt for a careful reading that led to some improvements.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to A. L’Moudden or È. Marchand.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

L’Moudden, A., Marchand, È. On Predictive Density Estimation under α-Divergence Loss. Math. Meth. Stat. 28, 127–143 (2019). https://doi.org/10.3103/S1066530719020030

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066530719020030

Keywords

AMS 2010 Subject Classification

Navigation