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Estimation of a parameter vector when some components are restricted

Author

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  • Fourdrinier, Dominique
  • Ouassou, Idir
  • Strawderman, William E.

Abstract

We consider the problem of estimating a p-dimensional parameter [theta]=([theta]1,...,[theta]p) when the observation is a p+k vector (X,U) where dim X=p and where U is a residual vector with dim U=k. The distributional assumption is that (X,U) has a spherically symmetric distribution around ([theta],0). Two restrictions on the parameter [theta] are considered. First we assume that [theta]i[greater-or-equal, slanted]0 for i=1,...,p and, secondly, we suppose that only a subset of these [theta]i are nonnegative. For these two settings, we give a class of estimators [delta](X,U)=[delta]0(X)+g(X)U'U which dominate, under the usual quadratic loss, a natural estimator [delta]0(X) which corresponds to the MLE in the normal case. Lastly, we deal with the situation where the parameter [theta] belongs to a cone of . We show that, under suitable condition, domination of the natural estimator adapted to this problem can be extended to a larger cone containing and to any orthogonal transformation of this cone.

Suggested Citation

  • Fourdrinier, Dominique & Ouassou, Idir & Strawderman, William E., 2003. "Estimation of a parameter vector when some components are restricted," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 14-27, July.
    • Handle: RePEc:eee:jmvana:v:86:y:2003:i:1:p:14-27
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    References listed on IDEAS

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    1. Ouassou, Idir & Strawderman, William E., 2002. "Estimation of a parameter vector restricted to a cone," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 121-129, January.
    2. Cellier, D. & Fourdrinier, D., 1995. "Shrinkage Estimators under Spherical Symmetry for the General Linear Model," Journal of Multivariate Analysis, Elsevier, vol. 52(2), pages 338-351, February.
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    1. Fourdrinier, Dominique & Strawderman, William E., 2008. "A unified and generalized set of shrinkage bounds on minimax Stein estimates," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2221-2233, November.
    2. Chang, Yuan-Tsung & Matsuda, Takeru & Strawderman, William E., 2019. "A note on improving on a vector of coordinate-wise estimators of non-negative means via shrinkage," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 143-150.
    3. Dominique Fourdrinier & William Strawderman & Martin Wells, 2006. "Estimation of a Location Parameter with Restrictions or “vague information” for Spherically Symmetric Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 73-92, March.
    4. Hisayuki Tsukuma, 2012. "Simultaneous estimation of restricted location parameters based on permutation and sign-change," Statistical Papers, Springer, vol. 53(4), pages 915-934, November.

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