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On improved estimators of the generalized variance

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  • Sinha, Bimal Kumar

Abstract

Treated in this paper is the problem of estimating with squared error loss the generalized variance [Sigma] from a Wishart random matrix S: p - p ~ Wp(n, [Sigma]) and an independent normal random matrix X: p - k ~ N([xi], [Sigma] [circle times operator] Ik) with [xi](p - k) unknown. Denote the columns of X by X(1) ,..., X(k) and set [psi](0)(S, X) = {(n - p + 2)!/(n + 2)!} S , [psi](i)(X, X) = min[[psi](i-1)(S, X), {(n - p + i + 2)!/(n + i + 2)!} S + X(1) X'(1) + ... + X(i) X'(i) ] and [Psi](i)(S, X) = min[[psi](0)(S, X), {(n - p + i + 2)!/(n + i + 2)!} S + X(1) X'(1) + ... + X(i) X'(i) ], i = 1,...,k. Our result is that the minimax, best affine equivariant estimator [psi](0)(S, X) is dominated by each of [Psi](i)(S, X), i = 1,...,k and for every i, [psi](i)(S, X) is better than [psi](i-1)(S, X). In particular, [psi](k)(S, X) = min[{(n - p + 2)!/(n + 2)!} S , {(n - p + 2)!/(n + 2)!} S + X(1)X'(1),..., {(n - p + k + 2)!/(n + k + 2)!} S + X(1)X'(1) + ... + X(k)X'(k)] dominates all other [psi]'s. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155-160 (1964)) on the estimation of the normal variance.

Suggested Citation

  • Sinha, Bimal Kumar, 1976. "On improved estimators of the generalized variance," Journal of Multivariate Analysis, Elsevier, vol. 6(4), pages 617-625, December.
  • Handle: RePEc:eee:jmvana:v:6:y:1976:i:4:p:617-625
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    Citations

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    Cited by:

    1. Tatsuya Kubokawa & M. S. Srivastava, 1999. ""Estimating the Covariance Matrix: A New Approach", June 1999," CIRJE F-Series CIRJE-F-52, CIRJE, Faculty of Economics, University of Tokyo.
    2. Kubokawa, T. & Srivastava, M. S., 2003. "Estimating the covariance matrix: a new approach," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 28-47, July.
    3. Misra, Neeraj & Singh, Harshinder & Demchuk, Eugene, 2005. "Estimation of the entropy of a multivariate normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 324-342, February.
    4. Iliopoulos, George & Kourouklis, Stavros, 1999. "Improving on the Best Affine Equivariant Estimator of the Ratio of Generalized Variances," Journal of Multivariate Analysis, Elsevier, vol. 68(2), pages 176-192, February.
    5. Sanat Sarkar, 1991. "Stein-type improvements of confidence intervals for the generalized variance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 43(2), pages 369-375, June.
    6. Tatsuya Kubokawa & M. S. Srivastava, 2002. "Estimating the Covariance Matrix: A New Approach," CIRJE F-Series CIRJE-F-162, CIRJE, Faculty of Economics, University of Tokyo.
    7. Iliopoulos, George, 2008. "UMVU estimation of the ratio of powers of normal generalized variances under correlation," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1051-1069, July.
    8. Elfessi, Abdulaziz, 1997. "Estimation of a linear function of the parameters of an exponential distribution from doubly censored samples," Statistics & Probability Letters, Elsevier, vol. 36(3), pages 251-259, December.

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