Existence conditions for the uniformly minimum risk unbiased estimators in a class of linear models. (English) Zbl 1032.62063

Summary: This paper studies the existence of uniformly minimum risk unbiased (UMRU) estimators of parameters in a class of linear models with an error vector having multivariate normal distribution or \(t\)-distribution, which includes the growth curve model, the extended growth curve model, the seemingly unrelated regression equations model, the variance components model, and so on. Necessary and sufficient existence conditions are established for UMRU estimators of the estimable linear functions of regression coefficients under convex losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model with normality assumption, the conclusions given in the literature can be derived by applying the general results in this paper. For the variance components model, the necessary and sufficient existence conditions are reduced as terse forms.


62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
Full Text: DOI


[1] S. Geisser, Growth curve analysis, in: P.R. Krishnaih (Ed.), Handbook of Statistics, Vol. 1, North-Holland, Amsterdam, pp. 89-115. · Zbl 0483.62038
[2] Kubokawa, T., Double shrinkage estimation of common coefficients in two regression equations with heteroscedasticity, J. multivariate anal., 67, 169-189, (1998) · Zbl 0953.62008
[3] Lehmann, E.L., Theory of point estimation, (1983), Wiley New York · Zbl 0522.62020
[4] Potthoff, R.F.; Roy, S.N., A generalized multivariate analysis of variance model useful especially for growth curve problem, Biometrika, 51, 313-326, (1964) · Zbl 0138.14306
[5] Rao, C.R., Estimation of parameters in a linear model, Ann. statist., 4, 1023-1037, (1976) · Zbl 0336.62055
[6] Srivastava, V.K.; Giles, D.E.A., Seemingly unrelated regression equations models, (1987), Marcel Dekker New York · Zbl 0568.62066
[7] Sutradhar, B.C.; Ali, M.M., Estimation of the parameters of regression model with a multivariate t error variable, Commun. statist. theory methods, 15, 429-450, (1986) · Zbl 0608.62061
[8] Verbyla, A.P.; Venables, W.N., An extension of the growth curve model, Biometrika, 75, 129-138, (1988) · Zbl 0636.62073
[9] D. Von Roson, Maximum likelihood estimates in multivariate linear normal models with special references to the growth curve model, Research Report 135, Department of Mathematics and Statistics, University of Stockholm, Stockholm, Sweden, 1984.
[10] Von Roson, D., Maximum likelihood estimators in multivariate linear normal model, J. multivariate anal., 31, 187-200, (1989) · Zbl 0686.62037
[11] Von Roson, D., Uniqueness conditions for maximum likelihood estimators in a multivariate linear model, J. statist. plann. inference, 36, 269-276, (1993)
[12] Wu, Q.G., Necessary and sufficient conditions for the existence of the UMRU estimators in growth curve models, Chinese sci. bull., 39, 89-92, (1994) · Zbl 0795.62054
[13] Wu, Q.G., Existence of the uniformly minimum risk unbiased estimator in seemingly unrelated regression system, Acta math. sinica (N.S.), 11, 23-28, (1995) · Zbl 0832.62051
[14] Wu, Q.G., Existence of the UMRU estimator in SURE model with special covariance structures, Chinese sci. bull., 42, 1149-1151, (1997) · Zbl 0942.62508
[15] Wu, Q.G., Existence conditions of the uniformly minimum risk unbiased estimators in extended growth curve models, J. statist. plann. inference, 69, 101-114, (1998) · Zbl 0924.62057
[16] Zellner, A., An efficient method of estimating seemingly unrelated regression and tests for aggregation bias, J. amer. statist. assoc., 57, 348-368, (1962) · Zbl 0113.34902
[17] Zellner, A., Bayesian and non-Bayesian analysis of the regression model with multivariate student-t error terms, J. amer. statist. assoc., 71, 400-405, (1976) · Zbl 0348.62026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.