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The general mixed linear model can be written y = Xβ + Zu + e, where β is a vector of fixed effects, u is a vector of random effects and e is a vector of random errors. In this note, we mainly aim at investigating the general necessary and sufficient conditions under which the best linear unbiased estimator for $${\varvec \varrho}({\varvec l}, {\varvec m}) = {\varvec l}{\varvec '}{\varvec \beta}+{\varvec m}{\varvec '}{\varvec u}$$ is also optimal under the misspecified model. In addition, we offer approximate conclusions in some special situations including a random regression model. [ABSTRACT FROM AUTHOR]

In the paper the problem of testing of two-sided hypotheses for variance components in mixed linear models is considered. When the uniformly most powerful invariant test does not exist (see e.g. Das and Sinha, in Proceedings of the second international Tampere conference in statistics, 1987; Gnot and Michalski, in Statistics 25:213-223, 1994; Michalski and Zmyślony, in Statistics 27:297-310, 1996) then to conduct the optimal statistical inference on model parameters a construction of a test with locally best properties is desirable, cf. Michalski (in Tatra Mountains Mathematical Publications 26:1-21, 2003). The main goal of this article is the construction of the locally best invariant unbiased test for a single variance component (or for a ratio of variance components). The result has been obtained utilizing Andersson's and Wijsman's approach connected with a representation of density function of maximal invariant (Andersson, in Ann Star 10:955-961, 1982; Wijsman, in Proceedings of fifth Berk Symp Math Statist Prob 1:389-400, 1967; Wijsman, in Sankhyā A 48:1-42, 1986; Khuri et al., in Statistical tests for mixed linear models. 1998) and from generalized Neyman-Pearson Lemma (Dantzig and Wald, in Ann Math Stat 22:87-93, 1951; Rao, in Linear statistical inference and its applications, 1973). One selected real example of an unbalanced mixed linear model is given, for which the power functions of the LBIU test and Wald's test (the F-test in ANOVA model) are computed, and compared with the attainable upper bound of power obtained by using Neyman-Pearson Lemma. [ABSTRACT FROM AUTHOR]

Gnot et al. (J Statist Plann Inference 30(I):223 236, 1992) have presented the formulae for computing Bayes invariant quadratic estimators of variance components in normal mixed linear models of the form N{y, X/β, ∑ki=1 δ2iVi}, where the matrices Vi, 1 ≤i≤k-1, are symmetric and nonnegative definite and Vk is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV1 M,...,MVk-1 M, M, where M is an orthogonal projector on the null space of X'. In the paper we discuss methods of construction of such a basis. We survey Malley's algorithms for finding the smallest quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms. We also consider a class of matrices sharing some of the symmetries common to MV1M,...,MVk-1 M, M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving the efficiency of Malley's algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices MV1M,...,MVk-1M, M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods. [ABSTRACT FROM AUTHOR]