@phdthesis{Heindl2022, author = {Heindl, Thomas}, title = {Cumulated sum processes of residuals for goodness-of-fit tests in linear regression models}, doi = {10.17904/ku.opus-773}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:824-opus4-7731}, school = {Katholische Universit{\"a}t Eichst{\"a}tt-Ingolstadt}, pages = {XVII, 206 Seiten : Diagramme}, year = {2022}, abstract = {Linear regression models are very popular among users as they allow an intuitive interpretation of the dependencies between an outcome variable and explanatory variables. However, to avoid false conclusions from such models, any statistical inference should be backed up by a goodness-of-fit test. These tests check whether the hypothesised class of possible regression functions has been adequately chosen. In this work, goodness-of-fit tests for univariate linear regression models based on the asymptotic distribution of residual CUSUM processes are studied. In the literature, a distinction is made between random designs and fixed designs when studying such goodness-of-fit tests. We prove that for each of these two scenarios (random and fixed design) there is a corresponding uniform design, we examine similarities and differences between these two scenarios, and prove that, when it comes to goodness-of-fit tests in in linear regression models based on the asymptotic distribution of residual CUSUM processes, one can assume without loss of generality a generic linear regression model. Furthermore, a geometric interpretation of residual CUSUM limit processes as projections on certain reproducing kernel Hilbert spaces is given, the effect of heteroscedastic regression errors on residual CUSUM limit processes is investigated, and an interpretation of the Khmaladze maratingale transformation as a continuous-time recursive least squares method is made explicit.}, subject = {Anpassungstest}, language = {en} }