Methods and algorithms for computing probability distributions of estimators and test statistics using characteristic functions

PhD study program: Applied Mathematics
Akademic year: 2024-2025
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:

Methods and algorithms will be developed in this dissertation to calculate the probability distributions of selected estimators and test statistics, as well as their applications. The focus will be on estimators (BLUE) and predictors (BLUP) in linear mixed models, particularly when there is a violation of standard assumptions regarding the normality distribution of errors and random effects. By assuming knowledge of the input variables’ distribution or utilizing the observed empirical distribution, numerical methods that rely on the inversion of the characteristic function of the estimators and test statistics under consideration can be employed. The main objective of this dissertation is to devise efficient numerical methods and algorithms for computing the probability distribution of functions of random variables, which can be determined using characteristic functions. This encompasses a wide range of distributions that have practical applications in natural, technical, and biomedical sciences. The research is connected to two APVV projects, namely “Advanced mathematical and statistical methods for measurement and metrology” and “Effective computational methods for the characterization of materials at the nano scale,” respectively. These projects will provide opportunities for international collaboration and active participation in significant domestic and foreign conferences. The completion of the dissertation will take place at the Institute of Measurement Science of the Slovak Academy of Sciences in Bratislava, in partnership with an external educational institution (EVI).

The aim of the dissertation is the development of methods and algorithms for the calculation of probability distributions of selected estimators and test statistics in models where there is a violation of standard assumptions about the normality of the distribution of errors.

The work is related to the solution of the scientific projects of APVV, MATHMER – “Advanced mathematical and statistical methods for measurement and metrology” and ECMeNaM “Effective computational methods for the characterization of materials at the nano scale”, which will enable the doctoral student to actively cooperate internationally and actively participate in important domestic and foreign conferences. The dissertation will be completed in a partner external educational institution (EVI), at the Institute of Measurement Science of the Slovak Academy of Sciences in Bratislava.

Literature:

WITKOVSKÝ, Viktor. Numerical inversion of a characteristic function: An alternative tool to form the probability distribution of output quantity in linear measurement models. In Acta IMEKO, 2016, vol. 5, no. 3, p. 32-44.
WITKOVSKÝ, V.: Estimation, testing, and prediction regions of the fixed and random effects by solving the Henderson’s mixed model equations. Measurement Science Review 12(6), 2012, 234-248.
WITKOVSKÝ, Viktor. CharFunTool: The characteristic functions toolbox, 2022. https://github.com/witkovsky/CharFunTool