Discussiones Mathematicae Probability and Statistics 22(1,2) (2002) 61-71


Roman Zmyślony and Stefan Zontek

Institute of Mathematics
University of Zielona Góra
Podgórna 50, 65-246 Zielona Góra, Poland


It is shown that a method of robust estimation in a two way crossed classification mixed model, recently proposed by Bednarski and Zontek (1996), can be extended to a more general case of variance components model with commutative a covariance matrices.

Keywords: Robust estimator, maximum likelihood estimator, statistical functional, Fisher consistency, Fréchet differentiability.

2000 Mathematics Subject Classification: 62F35, 62J05.


T. Bednarski, Fréchet differentiability and robust estimation, Asymptotic Statistics. Proc. of the Fifth Prague Symp. Physica Verlag, Springer, (1994), 49-58.
T. Bednarski, B.R. Clarke and W. Kokiewicz, Statistical expansions and locally uniform Fréchet differentiability, J. Australian Math. Soc., Ser. A 50 (1991), 88-97.
T. Bednarski and B.R. Clarke, Trimmed likelihood estimation of location and scale of the normal distribution, Australian J. Statists. 35 (1993), 141-153.
T. Bednarski and Z. Zontek, Robust estimation of parameters in mixed unbalanced models, Ann. Statist. 24 (4) (1996), 1493-1510.
B.R. Clarke, Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations, Ann. Statist. 11 (1983), 1196-1206.
B.R. Clarke, Nonsmooth analysis and Fréchet differentiability of M-functionals, Probab. Th. Rel. Fields 73 (1986), 197-209.
B. Iglewicz, Robust scale estimators and confidence intervals for location, D.C. Hoagling, F. Mosteller and J.W. Tukey, Eds., Understanding Robust and Exploratory Data Analysis, Wiley, New York, (1983), 404-431.
J. Kiefer, On large deviations of the empiric D.F. of vector chance variables and a law of iterated logarithm, Pacific J. Math. 11 (1961), 649-660.
D.M. Rocke, Robustness and balance in the mixed model, Biometrics 47 (1991), 303-309.
J. Seely, Quadratic subspaces and completness, Ann. Math. Statist. 42 (1971), 710-721.
R. Zmyślony and H. Drygas, Jordan algebras and Bayesian quadratic estimation of variance components, Linear Algebra and its Applications 168 (1992), 259-275.

Received 26 November 2002