Discussiones Mathematicae Probability and Statistics 24(1) (2004) 109-126

ON SOME PROPERTIES OF ML AND REML ESTIMATORS IN MIXED NORMAL MODELS WITH TWO VARIANCE COMPONENTS

Stanisław Gnot

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

Andrzej Michalski

Department of Mathematics
Agriculture University of Wrocław
Grunwaldzka 53, 50-357 Wrocław, Poland

Agnieszka Urbańska-Motyka

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

Abstract

In the paper, the problem of estimation of variance components s12 and s22 by using the ML-method and REML-method in a normal mixed linear model N{ Y, E(Y) = Xb,  Cov(Y) = s12 V+s22In } is considered. This paper deal with properties of estimators of variance components, particularly when an explicit form of these estimators is unknown. The conditions when the ML and REML estimators can be expressed in explicit forms are given, too. The simulation study for one-way classification unbalanced random model together with a new proposition of approximation of expectation and variances of ML and REML estimators are shown. Numerical calculations with reference to the generalized Fisher's information are also given.

Keywords: mixed linear models, likelihood-based inference, ML- and REML- estimation, variance components, Fisher's information.

2000 Mathematics Subject Classification: 62F10, 62F12.

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Received 3 December 2003