Flexibility, Robustness and Discontinuities in Nonparametric Regression Approaches

Thesis title: Flexibility, Robustness and Discontinuity in Nonparametric Regression Approaches Author: Mgr. Matúš Maciak, M.Sc. Department: Department of Probability and Mathematical Statistics, Charles University in Prague Supervisor: Prof. RNDr. Marie Hušková, DrSc. huskova@karlin.mff.cuni.cz Abstract: In this thesis we focus on local polynomial estimation approaches of an unknown regression function while taking into account also some robust issues like a presence of outlying observa- tions or heavy-tailed distributions of random errors as well. We will discuss the most common method used for such settings, so called local polynomial M-smoothers and we will present the main statistical properties and asymptotic inference for this method. The M-smoothers method is especially suitable for such cases because of its natural robust flavour, which can nicely deal with outliers as well as heavy-tailed distributed random errors. Another important quality we will focus in this thesis on is a discontinuity issue where we allow for sudden changes (discontinuity points) in the unknown regression function or its derivatives respectively. We will propose a discontinuity model with different variability structures for both independent and dependent random errors while the discontinuity points will be treated in a..