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Improved Nonnegative Estimation of Variance Components in Balanced Multivariate Mixed Models

Author

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  • Mathew, T.
  • Niyogi, A.
  • Sinha, B. K.

Abstract

Consider the independent Wishart matrices S1 W([Sigma] + [lambda][Theta],q1) and S2 W([Sigma], q2) where [Sigma] is an unknown positive definite (p.d.) matrix, [Theta] is an unknown nonnegative definite (n.n.d.) matrix, and [lambda] is a known positive scalar. For the estimation of [Theta], a class of estimators of the form [Theta](c,[epsilon]) = (c/[lambda]){S1/q1 - [epsilon](S2/q2)} (c >= 0, [epsilon] 0, [epsilon] > 0, the estimator obtained by taking the positive part of [Theta](c, [epsilon]) results in an n.n.d. estimator, say [Theta](c, [epsilon]) +, that is uniformly better than [Theta]U. Numerical results indicate that in terms of mean squared error, [Theta](c, [epsilon]) + performs much better than both [Theta]U and the restricted maximum likelihood estimator [Theta]REML of [Theta]. Similar results are also obtained for the nonnegative estimation of tr [Theta] and a'[Theta]a, where a is an arbitrary nonzero vector. For estimating [Sigma], we have derived estimators that are claimed to be uniformly better than the unbiased estimator [Sigma]U = S2/q2 under the squared error loss function and the entropy loss function. We have been able to establish the claim only in the bivariate case. Numerical results are reported showing the risk improvement of our proposed estimators of [Sigma].

Suggested Citation

  • Mathew, T. & Niyogi, A. & Sinha, B. K., 1994. "Improved Nonnegative Estimation of Variance Components in Balanced Multivariate Mixed Models," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 83-101, October.
    • Handle: RePEc:eee:jmvana:v:51:y:1994:i:1:p:83-101
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    Cited by:

    1. Oualkacha Karim & Labbe Aurelie & Ciampi Antonio & Roy Marc-Andre & Maziade Michel, 2012. "Principal Components of Heritability for High Dimension Quantitative Traits and General Pedigrees," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(2), pages 1-27, January.
    2. Hisayuki Tsukuma & Tatsuya Kubokawa, 2014. "Unified Improvements in Estimation of a Normal Covariance Matrix in High and Low Dimesions," CIRJE F-Series CIRJE-F-937, CIRJE, Faculty of Economics, University of Tokyo.
    3. Wu, Xiaoyong & Zou, Guohua & Li, Yingfu, 2009. "Uniformly minimum variance nonnegative quadratic unbiased estimation in a generalized growth curve model," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 1061-1072, May.
    4. Aryal, Subhash & Bhaumik, Dulal K. & Mathew, Thomas & Gibbons, Robert D., 2014. "An optimal test for variance components of multivariate mixed-effects linear models," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 166-178.
    5. Kubokawa, Tatsuya & Tsai, Ming-Tien, 2006. "Estimation of covariance matrices in fixed and mixed effects linear models," Journal of Multivariate Analysis, Elsevier, vol. 97(10), pages 2242-2261, November.
    6. Jian Zhao & Thomas Mathew, 2018. "Some Point Estimates and Confidence Regions for Multivariate Inter-laboratory Data Analysis," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 147-166, December.

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