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On improved estimators of the generalized variance


  • Sinha, Bimal Kumar


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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    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. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    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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