Methods and algorithms for exact probability distribution of the selected estimators and test statistics in linear mixed models and their applications

PhD study program: Applied Mathematics
Akademic year: 2021-2022
Advisor: doc. RNDr. Viktor Witkovský, CSc. (witkovsky@savba.sk)
External educational institution: Institute of Measurement Science SAS
Accepting university: Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Department of Applied Mathematics and Statistics

Annotation:

The aim of the dissertation is the development of methods and algorithms for calculating the probability distributions of estimators (BLUE) and predictors (BLUP) and selected test statistics in linear mixed models in case of violation of standard assumptions about the normality distribution of errors and random effects. Assuming knowledge of the distribution of input variables and their independence, it will be possible to use methods based on numerical inversion of the characteristic function of the considered estimates and test statistics. The specific goal of the dissertation will be the characterization of probability distributions expressible using G-functions and H-functions and the development of efficient numerical methods and algorithms for calculating the probability distribution for algebraic functions of independent random variables that can be expressed using these distributions. It turns out that the probability density function (PDF) resp. the cumulative distribution function (CDF) of many probability distributions can be expressed using these special functions (e.g., gamma distribution, beta distribution, chi-square distribution, F-distribution). It is a very wide class of distributions that find applications in various fields of natural, technical and biomedical sciences.

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