## Electronic Journal of Statistics

### On improved predictive density estimation with parametric constraints

#### Abstract

We consider the problem of predictive density estimation for normal models under Kullback-Leibler loss (KL loss) when the parameter space is constrained to a convex set. More particularly, we assume that $X\sim {\cal N}_{p}(\mu,v_{x}I)$ is observed and that we wish to estimate the density of $Y\sim {\cal N}_{p}(\mu,v_{y}I)$ under KL loss when μ is restricted to the convex set Cp. We show that the best unrestricted invariant predictive density estimator U is dominated by the Bayes estimator πC associated to the uniform prior πC on C. We also study so called plug-in estimators, giving conditions under which domination of one estimator of the mean vector μ over another under the usual quadratic loss, translates into a domination result for certain corresponding plug-in density estimators under KL loss. Risk comparisons and domination results are also made for comparisons of plug-in estimators and Bayes predictive density estimators. Additionally, minimaxity and domination results are given for the cases where: (i) C is a cone, and (ii) C is a ball.

#### Article information

Source
Electron. J. Statist., Volume 5 (2011), 172-191.

Dates
First available in Project Euclid: 14 April 2011

https://projecteuclid.org/euclid.ejs/1302784852

Digital Object Identifier
doi:10.1214/11-EJS603

Mathematical Reviews number (MathSciNet)
MR2792550

Zentralblatt MATH identifier
1274.62079

#### Citation

Fourdrinier, Dominique; Marchand, Éric; Righi, Ali; Strawderman, William E. On improved predictive density estimation with parametric constraints. Electron. J. Statist. 5 (2011), 172--191. doi:10.1214/11-EJS603. https://projecteuclid.org/euclid.ejs/1302784852

#### References

• [1] Aitchison, J. (1975) Goodness of prediction fit., Biometrika, 62 547–554.
• [2] Berry, C. (1990) Minimax estimation of a bounded normal mean vector., Journal of Multivariate Analysis, 35 130–139.
• [3] Brown, L. D., George, E. I. and Xu, X. (2008) Admissible predictive density estimation., Annals of Statistics, 36 1156–1170.
• [4] Casella, G. and Strawderman, W. E. (1981) Estimating a bounded normal mean., Annals of Statistics, 9 870–878.
• [5] Fourdrinier, D. and Marchand, É. (2010) On Bayes estimators with uniform priors on spheres and their comparative performance with maximum likelihood estimators for estimating bounded multivariate normal means., Journal of Multivariate Analysis, 101 1390–1399.
• [6] George, E. I., Feng, L. and Xu, X. (2006) Improved minimax predictive densities under Kullback-Leibler loss., Annals of Statistics, 34 78–91.
• [7] George, E. I. and Xu, X. (2010) Bayesian predictive density estimation., Frontiers of Statistical Decision Making and Bayesian Analysis. In honor of James O. Berger, 83–95. Springer.
• [8] Hartigan, J. A. (2004) Uniform priors on convex sets improve risk., Statistics and Probability Letters, 67 285–288.
• [9] Kiefer, J. (1957) Invariance, minimax sequential estimation, and continuous time processes., Annals of Mathematical Statistics, 28 573–601.
• [10] Komaki, F. (2001) A shrinkage predictive distribution for multivariate normal observables., Biometrika, 88 859–864.
• [11] Kubokawa, T. (2005) Estimation of bounded location and scale parameters., Journal of the Japanese Statistical Society, 35 221–249.
• [12] Liang, F. and Barron, A. (2004) Exact minimax strategies for predictive density estimation, data compression and model selection., IEEE Transactions on Information Theory, 50 2708–2726.
• [13] Liang, R. (1986) A note on the measurability of convex sets., Arch. Math, 47 90–92.
• [14] Marchand,É and Perron, F. (2001) Improving on the mle of a bounded normal mean., Annals of Statistics, 29 1066–1081.
• [15] Marchand,É and Perron, F. (2002) On the minimax estimator of a bounded normal mean., Statistics and Probability Letters, 58 327–333.
• [16] Marchand,É and Strawderman, W. E. (2010) A unified minimax result for restricted parameter spaces., Bernoulli, to appear.
• [17] Murray, G. D. (1977) A note on the estimation of probability density functions., Biometrika, 64 150–152.
• [18] Ng, V. M. (1980) On the estimation of parametric density functions., Biometrika, 67 505–506.
• [19] Rockafellar, R. T. (1996), Convex Analysis. Princeton University Press.
• [20] Tsukuma, H. and Kubokawa, T. (2008) Stein’s phenomenon in estimation of means restricted to a polyhedral convex cone., Journal of Multivariate Analysis, 99 141–164.