On locally optimal invariant unbiased tests for the variance components ratio in mixed linear models |
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Authors: | Andrzej Michalski |
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Institution: | (1) Institute of Plant Breeding, Seed Science and Population Genetics, University of Hohenheim, 70593 Stuttgart, Germany;(2) Bioinformatics Unit, Institute for Crop Production and Grassland Research, University of Hohenheim, 70599 Stuttgart, Germany;(3) HYBRO Saatzucht GmbH & Co. KG, Kleptow Nr. 53, 17291 Schenkenberg, Germany;(4) Institute of Agronomy and Plant Breeding II, Justus-Liebig-University Giessen, 35392 Giessen, Germany; |
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Abstract: | 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 Stat 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. |
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