Abstract
In linear models with two variance components, mean squared errors (MSEs) of invariant quadratic estimators of linear combinations of the variance components are quadratics in the variance components. However, it is revealing to note that MSEs are linear in convex combinations of the squares and products of the variance components, so that surfaces of MSEs are subsets of planes. Lower bounds on MSEs form a concave surface. MSEs of admissible invariant quadratic estimators are tangent to this surface, while MSEs of inadmissible estimators are not. There are different bounds when attention is restricted to unbiased invariant quadratic estimators or the more general class obtained by dropping the restriction to unbiasedness. The purpose of this paper is to develop these simple relations and to illustrate them with graphs.
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References
LaMotte, L. (1973). Quadratic estimation of variance components. Biometrics 29 311–330.
LaMotte, L. (1977). On admissibility and completeness of linear unbiased estimators in a general linear model. Journal of the American Statistical Association 72 438–441.
LaMotte, L. (1980). Some results on biased estimation applied to variance component estimation. In: W. Klonecki, A. Kozek and J. Rosiriski, Eds., Mathematical Statistics and Probability Theory. Lecture Notes in Statistics 2. Springer-Verlag, New York, 266–274.
LaMotte, L. (1982). Admissibility in linear estimation. Annals of Statistics 10 245–255.
LaMotte, L. and McWhorter, A. (1978). An exact test for the presence of random walk coefficients in a linear regression model. Journal of the American Statistical Association 73 816–820
Olsen, A., Seely, J. and Birkes, D. (1976). Invariant quadratic estimation for two variance compo-nents. The Annals of Statistics 4 878–890.
Pukelsheim, F. (1976). Estimating variance components in linear models. Journal of Multivariate Analysis 6 626–629.
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© 1994 Springer Science+Business Media Dordrecht
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Lamotte, L.R. (1994). Geometrical Relations Among Variance Component Estimators. In: Caliński, T., Kala, R. (eds) Proceedings of the International Conference on Linear Statistical Inference LINSTAT ’93. Mathematics and Its Applications, vol 306. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1004-4_12
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DOI: https://doi.org/10.1007/978-94-011-1004-4_12
Publisher Name: Springer, Dordrecht
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