Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-30T17:25:22.470Z Has data issue: false hasContentIssue false
  • English
  • Français

PREDICTIVE DENSITY ESTIMATION FOR MULTIPLE REGRESSION

Published online by Cambridge University Press:  15 January 2008

Edward I. George  and
Xinyi Xu
Edward I. George
Affiliation:
The Wharton School, University of Pennsylvania
Xinyi Xu
Affiliation:
The Ohio State University
Get access Rights & Permissions [Opens in a new window]

Abstract

Suppose we observe X ∼ Nm(Aβ,σ2I) and would like to estimate the predictive density p(y|β) of a future Y ∼ Nn(Bβ,σ2I). Evaluating predictive estimates by Kullback–Leibler loss, we develop and evaluate Bayes procedures for this problem. We obtain general sufficient conditions for minimaxity and dominance of the “noninformative” uniform prior Bayes procedure. We extend these results to situations where only a subset of the predictors in A is thought to be potentially irrelevant. We then consider the more realistic situation where there is model uncertainty and this subset is unknown. For this situation we develop multiple shrinkage predictive estimators and obtain general minimaxity and dominance conditions. Finally, we provide an explicit example of a minimax multiple shrinkage predictive estimator based on scaled harmonic priors.We acknowledge Larry Brown, Feng Liang, Linda Zhao, and three referees for their helpful suggestions. This work was supported by various NSF grants, DMS-0605102 the most recent.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 2 , April 2008 , pp. 528 - 544
Copyright
© 2008 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aitchison, J. (1975) Goodness of prediction fit. Biometrika 62, 547–554.Google Scholar
Brown, L.D. (1971) Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Annals of Mathematical Statistics 42, 855–903.Google Scholar
Brown, L.D., E.I. George, & X. Xu (2007) Admissible predictive density estimation. Annals of Statistics, forthcoming.Google Scholar
George, E.I. (1986a) Minimax multiple shrinkage estimation. Annals of Statistics 14, 188–205.Google Scholar
George, E.I. (1986b) Combining minimax shrinkage estimators. Journal of the American Statistical Association 81, 437–445.Google Scholar
George, E.I. (1986c) A formal Bayes multiple shrinkage estimator. Communications in Statistics: Part A—Theory and Methods, special issue, Stein-Type Multivariate Estimation 15, 2099–2114.Google Scholar
George, E.I., F. Liang, & X. Xu (2006) Improved minimax prediction under Kullback-Leibler Loss. Annals of Statistics 34, 78–91.Google Scholar
Liang, F. (2002) Exact minimax procedures for predictive density estimation and data compression. Ph.D. Dissertation, Department of Statistics, Yale University.
Liang, F. & A. Barron (2004) Exact minimax strategies for predictive density estimation, data compression and model selection. IEEE Information Theory Transactions 50, 2708–2726.Google Scholar
Steele, J.M. (2001) Stochastic Calculus and Financial Applications. Springer-Verlag.
Stein, C. (1974) Estimation of the mean of a multivariate normal distribution. In J. Hajek (ed.), Proceedings of the Prague Symposium on Asymptotic Statistics, pp. 345–381. Universita Karlova.
Stein, C. (1981) Estimation of a multivariate normal mean. Annals of Statistics 9, 1135–1151.Google Scholar