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Improved estimation for elliptically symmetric distributions with unknown block diagonal covariance matrix

Author

Listed:
  • Fourdrinier Dominique
  • Strawderman William E.
  • Wells Martin T.

Abstract

Let X, U1, …, Un-1 be n random vectors in ℝp with joint density of the form f((X - θ)´∑-1(X - θ) + ∑n-1j = 1U´j∑-1Uj) where both θ∈ℝp and ∑ are unknown, the scale matrix ∑ being supposed structured as a diagonal matrix, that is, ∑= diag(∑1, …,∑b) where, for 1 ≤ i ≤ b, ∑i is a pi × pi matrix and ∑i = 1bpi = p. We consider the problem of the estimation of θ with the invariant loss (δ - θ)´∑-1(δ - θ) and propose estimators which dominate the usual estimator δ0(X) = X. These domination results hold simultaneously for the entire class of such distributions. The proof uses a generalization of integration by parts formulae by Stein and Haff. We also consider estimating ∑ under LS(∑^,∑) = tr(∑^∑-1) - log |∑^∑-1| - p and propose estimators that dominate the unbiased estimator ∑^UB = diag(S1, …, Sb)/(n - 1), where Si = ∑j = 1n - 1UijU´ij and dim Uji = pi, for 1 ≤ i ≤ b and 1 ≤ j ≤ n - 1. The subsequent development of expressions is analogous to the unbiased estimators of risk technique and, in fact, reduces to an unbiased estimator of risk in the normal case.

Suggested Citation

  • Fourdrinier Dominique & Strawderman William E. & Wells Martin T., 2009. "Improved estimation for elliptically symmetric distributions with unknown block diagonal covariance matrix," Statistics & Risk Modeling, De Gruyter, vol. 26(3), pages 203-217, April.
  • Handle: RePEc:bpj:strimo:v:26:y:2009:i:3:p:203-217:n:4
    DOI: 10.1524/stnd.2008.1002
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    References listed on IDEAS

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    1. Haff, L. R., 1979. "An identity for the Wishart distribution with applications," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 531-544, December.
    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.
    3. Haff L. R. & Berger James, 1983. "A Class Of Minimax Estimators Of A Normal Mean Vector For Arbitrary Quadratic Loss And Unknown Covariance Matrix," Statistics & Risk Modeling, De Gruyter, vol. 1(2), pages 105-130, February.
    4. Kubokawa, T. & Srivastava, M. S., 2001. "Robust Improvement in Estimation of a Mean Matrix in an Elliptically Contoured Distribution," Journal of Multivariate Analysis, Elsevier, vol. 76(1), pages 138-152, January.
    5. Srivastava, M. S. & Bilodeau, M., 1989. "Stein estimation under elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 28(2), pages 247-259, February.
    6. Fourdrinier, Dominique & Strawderman, William E. & Wells, Martin T., 2003. "Robust shrinkage estimation for elliptically symmetric distributions with unknown covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 24-39, April.
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