Bernoulli

  • Bernoulli
  • Volume 18, Number 2 (2012), 635-643.

A unified minimax result for restricted parameter spaces

Éric Marchand and William E. Strawderman

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Abstract

We provide a development that unifies, simplifies and extends considerably a number of minimax results in the restricted parameter space literature. Various applications follow, such as that of estimating location or scale parameters under a lower (or upper) bound restriction, location parameter vectors restricted to a polyhedral cone, scale parameters subject to restricted ratios or products, linear combinations of restricted location parameters, location parameters bounded to an interval with unknown scale, quantiles for location-scale families with parametric restrictions and restricted covariance matrices.

Article information

Source
Bernoulli, Volume 18, Number 2 (2012), 635-643.

Dates
First available in Project Euclid: 16 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1334580727

Digital Object Identifier
doi:10.3150/10-BEJ336

Mathematical Reviews number (MathSciNet)
MR2922464

Zentralblatt MATH identifier
1251.49024

Keywords
covariance matrices linear combinations location parameters minimax polyhedral cones quantiles restricted parameters scale parameters

Citation

Marchand, Éric; Strawderman, William E. A unified minimax result for restricted parameter spaces. Bernoulli 18 (2012), no. 2, 635--643. doi:10.3150/10-BEJ336. https://projecteuclid.org/euclid.bj/1334580727


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References

  • [1] Berger, J.O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. New York: Springer.
  • [2] Blumenthal, S. and Cohen, A. (1968). Estimation of two ordered translation parameters. Ann. Math. Statist. 39 517–530.
  • [3] Brown, L.D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Institute of Mathematical Statistics Lecture Notes—Monograph Series 9. Hayward, CA: IMS.
  • [4] Eaton, M.L. (1989). Group Invariance Applications in Statistics. NSF-CBMS Regional Conference Series in Probability and Statistics 1. Hayward, CA: IMS.
  • [5] Farrell, R.H. (1964). Estimators of a location parameter in the absolutely continuous case. Ann. Math. Statist. 35 949–998.
  • [6] Ferguson, T.S. (1967). Mathematical Statistics: A Decision Theoretic Approach. Probability and Mathematical Statistics 1. New York: Academic Press.
  • [7] Girshick, M.A. and Savage, L.J. (1951). Bayes and minimax estimates for quadratic loss functions. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability 1950 53–73. Berkeley and Los Angeles: Univ. California Press.
  • [8] Hartigan, J.A. (2004). Uniform priors on convex sets improve risk. Statist. Probab. Lett. 67 285–288.
  • [9] James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 361–379. Berkeley, CA: Univ. California Press.
  • [10] Katz, M.W. (1961). Admissible and minimax estimates of parameters in truncated spaces. Ann. Math. Statist. 32 136–142.
  • [11] Kiefer, J. (1957). Invariance, minimax sequential estimation, and continuous time processes. Ann. Math. Statist. 28 573–601.
  • [12] Kubokawa, T. (2004). Minimaxity in estimation of restricted parameters. J. Japan. Statist. Soc. 34 229–253.
  • [13] Kubokawa, T. (2005). Estimation of a mean of a normal distribution with a bounded coefficient of variation. Sankhyā 67 499–525.
  • [14] Kubokawa, T. (2010). Minimax estimation of linear combinations of restricted location parameters. CIRJE discussion paper. Available at www.cirje.e.u-tokyo.ac.jp/research/dp/2010/2010cf723.pdf.
  • [15] Kumar, S. and Sharma, D. (1988). Simultaneous estimation of ordered parameters. Comm. Statist. Theory Methods 17 4315–4336.
  • [16] Lehmann, E.L. and Casella, G. (1998). Theory of Point Estimation, 2nd ed. New York: Springer.
  • [17] Marchand, E. and Strawderman, W.E. (2004). Estimation in restricted parameter spaces: A review. In A Festschrift for Herman Rubin. Institute of Mathematical Statistics Lecture Notes—Monograph Series 45 21–44. Beachwood, OH: IMS.
  • [18] 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.
  • [19] Marchand, É. and Strawderman, W.E. (2005). On improving on the minimum risk equivariant estimator of a scale parameter under a lower-bound constraint. J. Statist. Plann. Inference 134 90–101.
  • [20] Robert, C.P. (2001). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, 2nd ed. New York: Springer.
  • [21] Tsukuma, H. and Kubokawa, T. (2008). Stein’s phenomenon in estimation of means restricted to a polyhedral convex cone. J. Multivariate Anal. 99 141–164.
  • [22] van Eeden, C. (2006). Restricted Parameter Space Estimation Problems: Admissibility and Minimaxity Properties. Lecture Notes in Statistics 188. New York: Springer.