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A multivariate approach for estimating the random effects variance component in one-way random effects model

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  • Sutradhar, Brajendra C.

Abstract

It is well known that the ANOVA estimator of the random effects variance component in one-way random effects model can assume negative values. It is also well known that nonnegative quadratic unbiased estimators do not exist for estimating the random effects variance component (LaMotte, 1973). LaMotte (1985) indicated the possibility that nonnegative invariant quadratic estimator of the random effects variance component uniformly better than the ANOVA estimator may exist for the balanced one-way random effects model. Mathew et al. (1992a) have shown that such estimator exists only when the number of treatments is 9 or less. As noted in Herbach (1959) (see also Thompson, 1962), a simple truncation of the ANOVA estimator at zero yields uniform improvement over the ANOVA estimator. The estimators suggested by Herbach and Thompson are, in fact, restricted maximum likelihood estimators, and they are nonquadratic by nature. In this paper, we discuss a multivariate technique which always yields positive estimate of the random effects variance component in one-way random effects model. The multivariate approach exploits the estimates of the eigenvalues of the covariance matrix of the model in estimating the variance components including the error variance. The resulting estimates are nonquadratic. The success of this multivariate approach depends on the precise estimation of the eigenvalues. Since there does not exist any unbiased estimation procedure in the small sample case for the estimation of the eigenvalues, we use a delete-d jackknife procedure to estimate them. This delete-d jackknife based multivariate approach yields better estimates (in terms of mean squared error) for the random effects variance component than the restricted maximum likelihood estimation as well as Chow and Shao's (1988) nonquadratic estimation approaches, which is shown through a simulation study for the cases with number of treatments up to 20.

Suggested Citation

  • Sutradhar, Brajendra C., 1997. "A multivariate approach for estimating the random effects variance component in one-way random effects model," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 333-339, May.
    • Handle: RePEc:eee:stapro:v:33:y:1997:i:4:p:333-339
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    References listed on IDEAS

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    1. Sutradhar, B. C. & Bartlett, R. F., 1990. "An empirical study of the estimation of eigenvalues in connection with non-negative estimation of variance components of a one-way random effect model," Computational Statistics & Data Analysis, Elsevier, vol. 9(1), pages 21-35, January.
    2. Mathew Thomas & Sinha Bimal Kumar & Sutradhar Brajendra C., 1992. "Improved Estimation Of Error Variance In General Balanced Mixed Models," Statistics & Risk Modeling, De Gruyter, vol. 10(3), pages 227-238, March.
    3. Chow, Shein-Chung & Shao, Jun, 1988. "A new procedure for the estimation of variance components," Statistics & Probability Letters, Elsevier, vol. 6(5), pages 349-355, April.
    4. Mathew, Thomas & Sinha, Bimal Kumar & Sutradhar, Brajendra C., 1992. "Nonnegative estimation of variance components in unbalanced mixed models with two variance components," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 77-101, July.
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