## Bernoulli

• Bernoulli
• Volume 23, Number 4B (2017), 3197-3212.

### On predictive density estimation for location families under integrated absolute error loss

#### Abstract

This paper is concerned with estimating a predictive density under integrated absolute error ($L_{1}$) loss. Based on a spherically symmetric observable $X\sim p_{X}(\Vert x-\mu\Vert^{2})$, $x,\mu \in \mathbb{R}^{d}$, we seek to estimate the (unimodal) density of $Y\sim q_{Y}(\Vert y-\mu \Vert^{2})$, $y\in \mathbb{R}^{d}$. We focus on the benchmark (and maximum likelihood for unimodal $p$) plug-in density estimator $q_{Y}(\Vert y-X\Vert^{2})$ and, for $d\geq 4$, we establish its inadmissibility, as well as provide plug-in density improvements, as measured by the frequentist risk taken with respect to $X$. Sharper results are obtained for the subclass of scale mixtures of normal distributions which include the normal case. The findings rely on the duality between the predictive density estimation problem with a point estimation problem of estimating $\mu$ under a loss which is a concave function of $\Vert \hat{\mu}-\mu\Vert^{2}$, Stein estimation results and techniques applicable to such losses, and further properties specific to scale mixtures of normal distributions. Finally, (i) we address univariate implications for cases where there exist parametric restrictions on $\mu$, and (ii) we show quite generally for logconcave $q_{Y}$ that improvements on the benchmark mle can always be found among the scale expanded predictive densities $\frac{1}{c}q_{Y}(\frac{(y-x)^{2}}{c^{2}})$, with $c-1$ positive but not too large.

#### Article information

Source
Bernoulli, Volume 23, Number 4B (2017), 3197-3212.

Dates
Revised: September 2015
First available in Project Euclid: 23 May 2017

https://projecteuclid.org/euclid.bj/1495505090

Digital Object Identifier
doi:10.3150/16-BEJ842

Mathematical Reviews number (MathSciNet)
MR3654804

Zentralblatt MATH identifier
1382.62011

#### Citation

Kubokawa, Tatsuya; Marchand, Éric; Strawderman, William E. On predictive density estimation for location families under integrated absolute error loss. Bernoulli 23 (2017), no. 4B, 3197--3212. doi:10.3150/16-BEJ842. https://projecteuclid.org/euclid.bj/1495505090

#### References

• [1] Aitchison, J. (1975). Goodness of prediction fit. Biometrika 62 547–554.
• [2] Andrews, D.F. and Mallows, C.L. (1974). Scale mixtures of normal distributions. J. Roy. Statist. Soc. Ser. B 36 99–102.
• [3] Baranchik, A.J. (1970). A family of minimax estimators of the mean of a multivariate normal distribution. Ann. Math. Stat. 41 642–645.
• [4] Berger, J. (1975). Minimax estimation of location vectors for a wide class of densities. Ann. Statist. 3 1318–1328.
• [5] Brandwein, A.C., Ralescu, S. and Strawderman, W.E. (1993). Shrinkage estimators of the location parameter for certain spherically symmetric distributions. Ann. Inst. Statist. Math. 45 551–565.
• [6] Brandwein, A.C. and Strawderman, W.E. (1980). Minimax estimation of location parameters for spherically symmetric distributions with concave loss. Ann. Statist. 8 279–284.
• [7] Brandwein, A.C. and Strawderman, W.E. (1991). Generalizations of James–Stein estimators under spherical symmetry. Ann. Statist. 19 1639–1650.
• [8] Brown, L.D., George, E.I. and Xu, X. (2008). Admissible predictive density estimation. Ann. Statist. 36 1156–1170.
• [9] DasGupta, A. and Lahiri, S.N. (2012). Density estimation in high and ultra high dimensions, regularization, and the $L_{1}$ asymptotics. In Contemporary Developments in Bayesian Analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman. Inst. Math. Stat. (IMS) Collect. 8 1–23. Beachwood, OH: IMS.
• [10] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation. The $L{_{1}}$ View. New York: Wiley.
• [11] Feller, W. (1966). An Introduction to Probability Theory and Its Application, 2nd ed. New York: Wiley.
• [12] Fourdrinier, D., Marchand, É., Righi, A. and Strawderman, W.E. (2011). On improved predictive density estimation with parametric constraints. Electron. J. Stat. 5 172–191.
• [13] George, E.I., Liang, F. and Xu, X. (2006). Improved minimax predictive densities under Kullback–Leibler loss. Ann. Statist. 34 78–91.
• [14] Kano, Y. (1994). Consistency property of elliptical probability density functions. J. Multivariate Anal. 51 139–147.
• [15] Komaki, F. (2001). A shrinkage predictive distribution for multivariate normal observables. Biometrika 88 859–864.
• [16] Kubokawa, T., Marchand, É. and Strawderman, W.E. (2015). On predictive density estimation for location families under integrated squared error loss. J. Multivariate Anal. 142 57–74.
• [17] Kubokawa, T., Marchand, É. and Strawderman, W.E. (2015). On improved shrinkage estimators for concave loss. Statist. Probab. Lett. 96 241–246.
• [18] Kubokawa, T. and Saleh, A.K.M.D.E. (1994). Estimation of location and scale parameters under order restrictions. J. Statist. Res. 28 41–51.
• [19] Marchand, É. and Strawderman, W.E. (2005). Improving on the minimum risk equivariant estimator for a location parameter which is constrained to an interval or a half-interval. Ann. Inst. Statist. Math. 57 129–143.
• [20] Muirhead, R.J. (2005). Aspects of Multivariate Statistical Theory. 2nd ed. New York: Wiley.
• [21] Nadarajah, S. (2003). The Kotz-type distribution with applications. Statistics 37 341–358.
• [22] Strawderman, W.E. (1974). Minimax estimation of location parameters for certain spherically symmetric distributions. J. Multivariate Anal. 4 255–264.
• [23] Weizman, M.S. (1970). Measures of overlap of income distributions of white and negro families in the United States. Technical Report 22, US Department of Commerce.