In this article we deal with testing the hypotheses of the so-called structured mean vector and the structure of a covariance matrix. For testing the above mentioned hypotheses Jordan algebra properties are used and tests based on best quadratic unbiased estimators (BQUE) are constructed. For convenience coordinate-free approach (see Kruskal (1968) and Drygas (1970)) is used as a tool for characterization of best unbiased estimators and testing hypotheses. To obtain the test for mean vector, linear function of mean vector with the standard inner product in null hypothesis is changed into equivalent hypothesis about some quadratic function of mean parameters (it is shown that both hypotheses are equivalent and testable). In both tests the idea of the positive and negative part of quadratic estimators is applied to get the test, statistics which have F distribution under the null hypothesis. Finally, power functions of the obtained tests are compared with other known tests like LRT or Roy test. For some set for parameters in the model the presented tests have greater power than the above mentioned tests. In the article we present new results of coordinate-free approach and an overview of existing results for estimation and testing hypotheses about BCS models.

coordinate-free approach, Jordan algebra, multivariate model, block compound symmetric covariance structure, best unbiased estimators, testing structure of mean vector, testing independence of block variables

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