Abstract
In this chapter, we will consider the problem of estimating a location vector which is constrained to lie in a convex subset of \(\mathbb {R}^P\). Estimators that are constrained to a set should be constrasted to the shrinkage estimators discussed in Sect.2.4.4 where one has “vague knowledge” that a location vector is in or near the specified set and consequently wishes to shrink toward the set but does not wish to restrict the estimator to lie in the set. Much of the chapter is devoted to one of two types of constraint sets, balls, and polyhedral cones. However, Sect.7.2 is devoted to general convex constraint sets and more particularly to a striking result of Hartigan (2004) which shows that in the normal case, the Bayes estimator of the mean with respect to the uniform prior over any convex set , \(\mathcal {C}\), dominates X for all \(\theta \in \mathcal {C}\) under the usual quadratic loss ∥δ − θ∥2.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The risks of δ α and δ MLE cannot match either, since a linear combination of these two distinct estimators would improve on δ α.
References
Berry C (1990) Minimax estimation of a bounded normal mean vector. J Multivar Anal 35:130–139
Bickel PJ (1981) Minimax estimation of the mean of a normal distribution when the parameter space is restricted. Ann Stat 9(6):1301–1309
Brown LD, Johnstone I, MacGibbon KB (1981) Variation diminishing transformations: a direct approach to total positivity and its statistical applications. J Am Stat Assoc 76:824–832
Casella G, Strawderman WE (1981) Estimating a bounded normal mean. Ann Stat 9:870–878
Donoho DL, Liu RC, MacGibbon KB (1990) Minimax risk over hyperrectangles, and implications. Ann Stat 80(3):1416–1437
Fourdrinier D, Marchand E (2010) On Bayes estimators with uniform priors on spheres and their comparative performance with maximum likelihood estimators for estimating bounded multivariate normal means. J Multivar Anal 101:1390–1399
Fourdrinier D, Strawderman, Wells MT (2006) Estimation of a location parameter with restrictions or “vague information” for spherically symmetric distributions. Ann Inst Stat Math 58:73–92
Hartigan JA (2004) Uniform priors on convex sets improve risk. Stat Probab Lett 67:285–288
Isawa M, Moritani Y (1997) A note on the admissibility of the maximum likelihood estimator for a bounded normal mean. Stat Probab Lett 32:99–105
Johnstone IM, MacGibbon KB (1992) Minimax estimation of a constrained Poisson vector. Ann Stat 20:807–831
Katz MW (1961) Admissible and minimax estimates of parameters in truncated spaces. Ann Math Stat 32(1):136–142
Kucerovsky D, Marchand E, Payandeh AT, Strawderman WE (2009) On the bayesianity of maximum likelihood estimators of restricted location parameters under absolute value error loss. Stat Decis Int Math J Stoch Methods Model 27:145–168
Levit BY (1981) On asymptotic minimax estimates of the second order. Theory Probab Its Appl 25(3):552–568
Marchand É, Perron F (2001) Improving on the MLE of a bounded normal mean. Ann Stat 29(4):1078–1093
Marchand É, Perron F (2002) On the minimax estimator of a bounded normal mean. Stat Probab Lett 58:327–333
Marchand É, Perron F (2005) Improving on the MLE of a bounded location parameter for spherical distributions. J Multivar Anal 92(2):227–238
Marchand É, Strawderman WE (2004) Estimation in restricted parameter spaces: a review. In: DasGupta A (ed) A Festschrift for Herman Rubin. Lecture notes–monograph series, vol 45. Institute of Mathematical Statistics, Beachwood, pp 21–44
Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York
Sengupta D, Sen PK (1991) Shrinkage estimation in a restricted parameter space. Sankhyā 53:389–411
Stoer J, Witzgall C (1970) Convexity and optimization in finite dimensions I, Die Grundlehren der mathematischen Wissenschaften, vol 163. Springer, Berlin
van Eeden C (2006) Restricted parameter space estimation problems: admissibility and minimaxity properties. Lecture notes in statistics, Springer, New York
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Fourdrinier, D., Strawderman, W.E., Wells, M.T. (2018). Restricted Parameter Spaces. In: Shrinkage Estimation. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-02185-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-02185-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02184-9
Online ISBN: 978-3-030-02185-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)