Skip to main content
SpringerLink
Log in

Improved estimation of the covariance matrix and the generalized variance of a multivariate normal distribution: some unifying results

  • Published:
Sankhya A Aims and scope Submit manuscript

Abstract

Suppose that there is a typical (scale equivariant) improved estimator of the variance, σ 2, of a univariate normal distribution with unknown mean at our disposal. Using this estimator, in this work we construct in a very simple way an improved estimator of the covariance matrix, Σ, of a multivariate normal distribution with unknown mean and another improved estimator of the generalized variance, \(\lvert \Sigma\rvert\). The data is a sample of i.i.d. observations from this distribution. The loss function is the entropy loss or the quadratic loss in the case of Σ and a general loss satisfying a certain condition in the case of \(\lvert\Sigma\rvert\). These novel results reduce, in a specific way, the problems of estimating Σ or \(\lvert\Sigma\rvert\) to the univariate problem of estimating σ 2. As a consequence, Stein-type, Brewster and Zidek-type, Strawderman-type and Maruyama-type improved estimators for Σ and \(\lvert\Sigma\rvert\) are directly constructed from their univariate counterparts. This work unifies and extends previously obtained results on the estimation of Σ and \(\lvert\Sigma\rvert\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bobotas, P., Iliopoulos, G. and Kourouklis, S. (2012). Estimating the ratio of two scale parameters: a simple approach. Ann. Inst. Statist. Math., 64, 343–357.

    Article  MathSciNet  MATH  Google Scholar 

  • Brewster, J.F. and Zidek, J.V. (1974). Improving on equivariant estimators. Ann. Statist., 2, 21–38.

    Article  MathSciNet  MATH  Google Scholar 

  • Brown, L.D. (1968). Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters. Ann. Math. Statist., 39, 29–48.

    Article  MathSciNet  MATH  Google Scholar 

  • Dey, D.K. and Srinivasan, C. (1985). Estimation of a covariance matrix under Stein’s loss. Ann. Statist., 13, 1581–1591.

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh, M. (1994). On Some Bayesian Solutions of the Neyman–Scott Problem. In Statistical Decision Theory and Related Topics, vol. V. Springer, New York, pp. 267–276.

    Chapter  Google Scholar 

  • Gupta, A.K. and Ofori-Nyarko, S. (1995). Improved minimax estimators of normal covariance and precision matrices. Statistics, 26, 19–25.

    Article  MathSciNet  MATH  Google Scholar 

  • Haff, L.R. (1980). Empirical Bayes estimation of the multivariate normal covariance matrix. Ann. Statist., 8, 586–597.

    Article  MathSciNet  MATH  Google Scholar 

  • Hara, H. (2001). Other classes of minimax estimators of variance covariance matrix in multivariate normal distribution. J. Multivariate Anal., 77, 175–186.

    Article  MathSciNet  MATH  Google Scholar 

  • Hara, H. (2002). Estimating a covariance matrix in MANOVA model. Statist. Decisions, 20, 257–267.

    MathSciNet  MATH  Google Scholar 

  • Iliopoulos, G. (1999). Point and Interval Estimation for Certain Scale Parameters. PhD Dissertation, Dept. of Mathematics, University of Patras (in Greek).

  • Iliopoulos, G. and Kourouklis, S. (1999). Improving on the best affine equivariant estimator of the ratio of generalized variances. J. Multivariate Anal., 68, 176–192.

    Article  MathSciNet  MATH  Google Scholar 

  • James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. Fourth Symp. Math. Statist. Prob, vol. 1, pp. 361–380. University of California Press.

  • Kubokawa, T. (1989). Improved estimation of a covariance matrix under quadratic loss. Statist. Probab. Lett., 8, 69–71.

    Article  MathSciNet  MATH  Google Scholar 

  • Kubokawa, T. (1994). A unified approach to improving equivariant estimators. Ann. Statist., 22, 290–299.

    Article  MathSciNet  MATH  Google Scholar 

  • Kubokawa, T. (1999). Shrinkage and modification techniques in estimation of variance and the related problems: a review. Comm. Statist. Theory Methods, 28, 613–650.

    Article  MathSciNet  MATH  Google Scholar 

  • Kubokawa, T., Roberts, C. and Saleh A.K.Md.E. (1992). Empirical Bayes estimation of the covariance matrix of a normal distribution with unknown mean under an entropy loss. Sankhyā A, 54, 402–410.

    MATH  Google Scholar 

  • Kubokawa, T., Honda, T., Morita, K. and Saleh A.K.Md.E. (1993). Estimating a covariance matrix of a normal distribution with unknown mean. J. Japan Statist. Soc., 23, 131–144.

    MathSciNet  MATH  Google Scholar 

  • Kubokawa, T. and Srivastava, M.S. (2003). Estimating the covariance matrix: a new approach. J. Multivariate Anal., 86, 28–47.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T., Jia, L. and Su, Y. (2012). A new estimator of covariance matrix. J. Statist. Plann. Inference, 142, 529–536.

    Article  MathSciNet  MATH  Google Scholar 

  • Maatta, J.M. and Casella, G. (1990). Developments in decision-theoretic variance estimation. Statist. Sci., 5, 90–101.

    MathSciNet  MATH  Google Scholar 

  • Maruyama, Y. (1998). Minimax estimators of a normal variance. Metrika, 48, 209–214.

    Article  MathSciNet  MATH  Google Scholar 

  • Maruyama, Y. and Strawderman, W.E. (2006). A new class of minimax generalized Bayes estimators of a normal variance. J. Statist. Plann. Inference, 136, 3822–3836.

    Article  MathSciNet  MATH  Google Scholar 

  • Pal, N. and Ling, C. (1995). Improved minimax estimation of powers of the variance of a multivariate normal distribution under the entropy loss function. Statist. Probab. Lett., 24, 205–211.

    Article  MathSciNet  MATH  Google Scholar 

  • Pal, N., Ling, C. and Lin, J. (1998). Estimation of a normal variance—a critical review. Statist. Papers, 4, 389–404.

    Article  MathSciNet  Google Scholar 

  • Perron, F. (1990). Equivariant estimators of the covariance matrix. Canad. J. Statist., 18, 179–182.

    Article  MathSciNet  MATH  Google Scholar 

  • Perron, F. (1992). Minimax estimators of a covariance matrix. J. Multivariate Anal., 43, 16–28.

    Article  MathSciNet  MATH  Google Scholar 

  • Petropoulos, C. and Kourouklis, S. (2005). Estimation of a scale parameter in mixture models with unknown location. J. Statist. Plann. Inference, 128, 191–218.

    Article  MathSciNet  MATH  Google Scholar 

  • Rukhin, A.L. and Ananda, M.M.A. (1992). Risk behavior of variance estimators in multivariate normal distribution. Statist. Probab. Lett., 13, 159–166.

    Article  MathSciNet  MATH  Google Scholar 

  • Shinozaki, N. (1995). Some modifications of improved estimators of a normal variance. Ann. Inst. Statist. Math., 47, 273–286.

    Article  MathSciNet  MATH  Google Scholar 

  • Shorrock, R.B. and Zidek, J.V. (1976). An improved estimator of the generalized variance. Ann. Statist., 4, 629–638.

    Article  MathSciNet  MATH  Google Scholar 

  • Sinha, B.K. (1976). On improved estimators of the generalized variance. J. Multivariate Anal., 6, 617–625.

    Article  MathSciNet  MATH  Google Scholar 

  • Sinha, B.K. and Ghosh, M. (1987). Inadmissibility of the best equivariant estimators of the variance-covariance matrix, the precision matrix and the generalized variance under entropy loss. Statist. Decisions, 5, 201–227.

    MathSciNet  MATH  Google Scholar 

  • Stein, C. (1964). Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean. Ann. Inst. Statist. Math., 16, 155–160.

    Article  MathSciNet  MATH  Google Scholar 

  • Strawderman, W.E. (1974). Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist., 2, 190–198.

    Article  MathSciNet  MATH  Google Scholar 

  • Sugiura, N. (1988). A class of improved estimators of powers of the generalized variance and precision under squared loss. In Statistical Theory and Data Analysis II (K. Matusita, ed.). North Holland, Amsterdam, pp 421–428.

  • Sugiura, N. (1989). Entropy loss and a class of improved estimators for powers of the generalized variance. Sankhyā A, 51, 328–333.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Patras, 26504, Patras, Greece

    Panayiotis Bobotas & Stavros Kourouklis

Authors
  1. Panayiotis Bobotas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stavros Kourouklis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stavros Kourouklis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bobotas, P., Kourouklis, S. Improved estimation of the covariance matrix and the generalized variance of a multivariate normal distribution: some unifying results. Sankhya A 75, 26–50 (2013). https://doi.org/10.1007/s13171-012-0021-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13171-012-0021-9

AMS (2000) subject classification.

Keywords and phrases.

Search

Navigation