Electronic Journal of Statistics

On predictive density estimation with additional information

Éric Marchand and Abdolnasser Sadeghkhani

Full-text: Open access

Abstract

Based on independently distributed $X_{1}\sim{N} _{p}(\theta _{1},\sigma ^{2}_{1}I_{p})$ and $X_{2}\sim{N}_{p}(\theta _{2},\sigma ^{2}_{2}I_{p})$, we consider the efficiency of various predictive density estimators for $Y_{1}\sim N_{p}(\theta _{1},\sigma ^{2}_{Y}I_{p})$, with the additional information $\theta _{1}-\theta _{2}\in A$ and known $\sigma ^{2}_{1},\sigma ^{2}_{2},\sigma ^{2}_{Y}$. We provide improvements on benchmark predictive densities such as those obtained by plug-in, by maximum likelihood, or as minimum risk equivariant. Dominance results are obtained for $\alpha -$divergence losses and include Bayesian improvements for Kullback-Leibler (KL) loss in the univariate case ($p=1$). An ensemble of techniques are exploited, including variance expansion, point estimation duality, and concave inequalities. Representations for Bayesian predictive densities, and in particular for $\hat{q}_{\pi_{U,A}}$ associated with a uniform prior for $\theta =(\theta _{1},\theta _{2})$ truncated to $\{\theta\in \mathbb{R}^{2p}:\theta _{1}-\theta _{2}\in A\}$, are established and are used for the Bayesian dominance findings. Finally and interestingly, these Bayesian predictive densities also relate to skew-normal distributions, as well as new forms of such distributions.

Article information

Source
Electron. J. Statist., Volume 12, Number 2 (2018), 4209-4238.

Dates
Received: January 2018
First available in Project Euclid: 15 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1544842900

Digital Object Identifier
doi:10.1214/18-EJS1493

Mathematical Reviews number (MathSciNet)
MR3892140

Keywords
Additional information $\alpha $-divergence loss bayes estimators dominance duality frequentist risk Kullback-Leibler loss multivariate normal plug-in predictive density restricted parameter skew-normal variance expansion

Rights
Creative Commons Attribution 4.0 International License.

Citation

Marchand, Éric; Sadeghkhani, Abdolnasser. On predictive density estimation with additional information. Electron. J. Statist. 12 (2018), no. 2, 4209--4238. doi:10.1214/18-EJS1493. https://projecteuclid.org/euclid.ejs/1544842900


Export citation

References

  • [1] Aitchison, J. (1975). Goodness of prediction fit., Biometrika, 62, 547-554.
  • [2] Aitchison, J. & Dunsmore, I.R., (1975)., Statistical Prediction Analysis. Cambridge University Press.
  • [3] Arellano-Valle, R.B., Branco, M.D., & Genton, M.G. (2006). A unified view on skewed distributions arising from selections., Canadian Journal of Statistics, 34, 581-601.
  • [4] Arnold, B.C. & Beaver, R.J. (2002). Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion)., Test, 11, 7-54.
  • [5] Arnold, B.C., Beaver, R.J., Groeneveld, R.A., & Meeker, W.Q. (1993). The nontruncated marginal of a truncated bivariate normal distribution., Psychometrika, 58, 471-488.
  • [6] Azzalini, A. (1985). A class of distributions which includes the normal ones., Scandinavian Journal of Statistics, 12, 171-178.
  • [7] Berry, C. (1990). Minimax estimation of a bounded normal mean vector., Journal of Multivariate Analysis, 35, 130-139.
  • [8] Blumenthal, S. & Cohen, A. (1968). Estimation of the larger translation parameter., Annals of Mathematical Statistics, 39, 502-516.
  • [9] Brandwein, A.C. & Strawderman, W.E. (1980). Minimax estimation of location parameters for spherically symmetric distributions with concave loss., Annals of Statistics, 8, 279-284.
  • [10] Brown, L.D., George, E.I., & Xu, X. (2008). Admissible predictive density estimation., Annals of Statistics, 36, 1156-1170.
  • [11] Brown, L.D. (1986)., Foundations of Exponential Families. IMS Lecture Notes, Monograph Series 9, Hayward, California.
  • [12] Cohen, A. & Sackrowitz, H.B. (1970). Estimation of the last mean of a monotone sequence., Annals of Mathematical Statistics, 41, 2021-2034.
  • [13] Corcuera, J.M. & Giummolè, F. (1999). A generalized Bayes rule for prediction. Scandinavian Journal of Statistics, 26, 265-279.
  • [14] Csiszàr, I. (1967). Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar., 2, 299-318.
  • [15] Dunson, D.B. & Neelon, B. (2003). Bayesian inference on order-constrained parameters in generalized linear models., Biometrics, 59, 286-295.
  • [16] Fourdrinier, D. & 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.
  • [17] Fourdrinier, D., Marchand, É., Righi, A. & Strawderman, W.E. (2011). On improved predictive density estimation with parametric constraints., Electronic Journal of Statistics, 5, 172-191.
  • [18] George, E.I., Liang, F. & Xu, X. (2006). Improved minimax predictive densities under Kullback-Leibler loss., Annals of Statistics, 34, 78-91.
  • [19] Ghosh, M., Mergel, V. & Datta, G.S. (2008). Estimation, prediction and the Stein phenomenon under divergence loss., Journal of Multivariate Analysis, 99, 1941-1961.
  • [20] Gupta, R.C. & Gupta, R.D. (2004). Generalized skew normal model., Test, 13, 501-524.
  • [21] Hartigan, J. (2004). Uniform priors on convex sets improve risk., Statistics & Probability Letters, 67, 285-288.
  • [22] Hwang, J.T.G. & Peddada, S.D. (1994). Confidence interval estimation subject to order restrictions., Annals of Statistics, 22, 67-93.
  • [23] Kato, K. (2009). Improved prediction for a multivariate normal distribution with unknown mean and variance., Annals of the Institute of Statistical Mathematics, 61, 531-542.
  • [24] Komaki, F. (2001). A shrinkage predictive distribution for multivariate normal observables., Biometrika, 88, 859-864.
  • [25] Kubokawa, T. (2005). Estimation of bounded location and scale parameters., Journal of the Japanese Statistical Society, 35, 221-249.
  • [26] Kubokawa, T., Marchand, É., & Strawderman, W.E. (2015). On predictive density estimation for location families under integrated squared error loss., Journal of Multivariate Analysis, 142, 57-74.
  • [27] Kubokawa, T., Marchand, É. & Strawderman, W.E. (2017). On predictive density estimation for location families under integrated absolute value loss., Bernoulli, 23, 3197-3212.
  • [28] Lee, C.I.C. (1981). The quadratic loss of isotonic regression under normality., Annals of Statistics, 9, 686-688.
  • [29] Liseo, B. & Loperfido, N. (2003). A Bayesian interpretation of the multivariate skew-normal distribution., Statistics & Probability Letters, 61, 395-401.
  • [30] Marchand, É., Jafari Jozani, M. & Tripathi, Y.M. (2012). On the inadmissibility of various estimators of normal quantiles and on applications to two-sample problems with additional information, Contemporary Developments in Bayesian analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman, Institute of Mathematical Statistics Volume Series, 8, 104-116.
  • [31] Marchand, É., Perron, F., & Yadegari, I. (2017). On estimating a bounded normal mean with applications to predictive density estimation., Electronic Journal of Statistics, 11, 2002-2025.
  • [32] Marchand, É. & Perron, F. (2001). Improving on the MLE of a bounded normal mean., Annals of Statistics, 29, 1078-1093.
  • [33] Marchand, É. & Strawderman, W.E. (2004). Estimation in restricted parameter spaces: A review., Festschrift for Herman Rubin, IMS Lecture Notes-Monograph Series, 45, 21-44.
  • [34] Maruyama, Y. & Strawderman, W.E. (2012). Bayesian predictive densities for linear regression models under $\alpha -$divergence loss: Some results and open problems., Contemporary Developments in Bayesian analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman, Institute of Mathematical Statistics Volume Series, 8, 42-56.
  • [35] Park, Y., Kalbfleisch, J.D. & Taylor, J. (2014). Confidence intervals under order restrictions., Statistica Sinica, 24, 429-445.
  • [36] Robert, C. (1996). Intrinsic loss functions., Theory and Decision, 40, 192-214.
  • [37] Robert, C. (1990). Modified Bessel functions and their applications in probability and statistics., Statistics and Probability Letters, 9, 155-161.
  • [38] Sadeghkhani, A. (2017). Estimation d’une densité prédictive avec information additionnelle. Ph.D. thesis. Université de Sherbrooke, (http://savoirs.usherbrooke.ca/handle/11143/11238)
  • [39] Shao, P.Y.-S. & Strawderman, W. (1996). Improving on the mle of a positive normal mean., Statistica Sinica, 6, 275-287.
  • [40] Silvapulle, M.J. & Sen, P.K. (2005)., Constrained statistical inference. Inequality, order, and shape restrictions. Wiley Series in Probability and Statistics. Wiley-Interscience.
  • [41] Spiring, F.A. (1993). The reflected normal loss function., Canadian Journal of Statistics, 31, 321-330.
  • [42] Stein, C. (1981). Estimation of the mean of a multivariate normal distribution., Annals of Statistics, 9, 1135-1151.
  • [43] van Eeden, C. & Zidek, J.V. (2001). Estimating one of two normal means when their difference is bounded., Statistics & Probability Letters, 51, 277-284.
  • [44] van Eeden, C. & Zidek, J.V. (2003). Combining sample information in estimating ordered normal means., Sankhy$\bara$ A, 64, 588-610.
  • [45] van Eeden, C. (2006)., Restricted parameter space problems: Admissibility and minimaxity properties. Lecture Notes in Statistics, 188, Springer.
  • [46] Zelazo, P.R., Zelazo, N.A. & Kolb, S. (1972). “Walking in the newborn”., Science, 176, 314-315.