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Submitted dissertation focuses on properties of multivariate generalized linear models, special attention is paid to powers of tests of linear hypothesis on parameters of these models. The dissertation consists of six chapters and three appendixes. Chapter 1 is dedicated to selected terms in the theory of maximum likelihood. Prepositions on properties of distribution from exponential family are considered in chapter 2. Assumptions assuring regularity of corresponding density function are derived among other things. Beside the properties of the most frequently studied multivariate distribution from exponential family, the multinomial distribution, properties of Wishart and transformed Dirichlet distribution are introduced. Afterwards, the multivariate generalized linear model (MGLM) is defined according to the monograph (Fahrmeir, 1994). Selected theorems on asymptotic properties of the model from paper (Fahrmeir, 1985) are included in chapter 3. Their proofs are performed in more details in contrast to the paper. Asymptotic confidence sets for vector of contrasts of parameters of MGLM are also derived. Finally, test statistics usually used for test of the linear hypothesis on parameters of MGLM are introduced in this chapter. Chapter 3 is concluded by application of model in genetic prediction of sepsis in children patients. Following chapters are dedicated to approximations of powers of the tests of linear hypothesis on parameters of MGLM. In chapter 4, attention is paid to approximations of powers of the tests of hypothesis against sequence of Pitman alternatives. Technique of these approximations is derived in appendix C in more details. Conversely, chapter 5, in which published results of the author of dissertation are generalized, deals with approximations of powers of tests in balanced one-way ANOVA type MGLM. Here, the powers of test of hypothesis against fixed alternative are considered in contrast to the previous chapter. Both approaches to approximations of the powers in balanced one-way ANOVA type MGLM are compared in chapter 6, especially by simulation studies in MGLM with multinomial, Wishart, transformed Dirichlet, binomial, Poisson, gamma and negative binomial distribution. Calculations presented in the dissertation were carried out using MATLAB programs, which are to be found on enclosed CD. References Fahrmeir, Ludwig, Kaufmann, Heinz. Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in Generalized Linear Models. The Annals of Statistics, Mar. 1985, Vol. 13, No. 1, s. 342-368. Fahrmeir, Ludwig, Tutz, Gerhard. Multivariate Statistical Modelling Based on Generalized Linear Models. New York: Springer - Verlag, 1994. ISBN 0387942335.
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