Abstract
This paper addresses the problem of estimating means of Hudson (1978) type exponential families, where the vector of means lies in a closed convex set with a piecewise smooth boundary. Instead of Stein (1981)-like integration-by-parts technique, the Gauss divergence theorem is used to provide an inequality for evaluation of the risk function with respect to a quadratic loss. The inequality shows that a James and Stein (1961) type estimator is superior to the least squares estimator subject to restriction on the closed convex set.
Similar content being viewed by others
References
Amirdjanova, A. and Woodroofe, M. (2004). Shrinkage estimation for convex poly-hedral cones. Statist. Probab. Lett., 70, 87–94.
Berger, J. (1980). Improving on inadmissible estimators in continuous exponential families with applications to simultaneous estimation of gamma scale parameters. Ann. Statist., 8, 545–571.
Chang, Y.-T. (1981). Stein-type estimators for parameters restricted by linear inequalities. Keio Sci. Tech. Rep., 34, 83–95.
Chang, Y.-T. (1982). Stein-type estimators of parameters in truncated spaces. Keio Sci. Tech. Rep., 35, 185–193.
Federer, H. (1969). Geometric Measure Theory. Springer-Verlag, New York.
Fleming, W. (1977). Functions of several variables, 2nd ed. Springer-Verlag, New York.
Fourdrinier, D., Ouassou, I. and Strawderman, W.E. (2003). Estimation of a parameter vector when some components are restricted. J. Multivariate Anal., 86, 14–27.
Haff, L.R. and Johnson, R.W. (1986). The superharmonic condition for simultaneous estimation of mean in exponential families. Canad. J. Statist., 14, 43–54.
Hudson, H.M. (1978). A natural identity for exponential families with application in multiparameter estimation. Ann. Statist., 6, 473–484.
James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. Fourth Berkeley Symp. Math. Statist. Probab., 1, 361–379. University of California Press, Berkeley.
Lehmann, E.L. and Casella, G. (1998). Theory of Point Estimation, 2nd ed. Springer-Verlag, New York.
Marchand, E. and Strawderman, W.E. (2004). Estimation in restricted parameter spaces: A review. In A Festschrift to Honor Herman Rubin (A. Dasgupta, ed.), IMS Lecture Notes-Monograph Series, Vol. 45, 21–44.
Oono, Y. and Shinozaki, N. (2006). Estimation of error variance in ANOVA model and order restricted scale parameters. Ann. Inst. Statist. Math., 58, 739–756.
Ouassou, I. and Strawderman, W.E. (2002). Estimation of a parameter vector restricted to a cone. Statist. Probab. Lett., 56, 121–129.
Robertson, T., Wright, F.T. and Dykstra, R. (1988). Order Restricted Statistical Inference. Wiley, Chichester.
Silvapulle, M.J. and Sen, P.K. (2005). Constrained Statistical Inference: Inequality, Order, and Shape Restrictions. Wiley, Hoboken.
Spivak, M. (1971). Calculus on Manifolds, 5th ed. Westview Press.
Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist., 9, 1135–1151.
Tsukuma, H. and Kubokawa, T. (2008). Minimax estimation of normal precisions via expansion estimators. to appear in J. Statist. Plan. Infer.
van Eeden, C. (2006). Restricted Parameter Space Estimation Problems. Springer-Verlag, New York.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tsukuma, H. Simultaneous estimation of restricted means via the gauss divergence theorem. Sankhya A 73, 110–124 (2011). https://doi.org/10.1007/s13171-011-0001-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-011-0001-5