@phdthesis{Evers2022, author = {Evers, Karsten}, title = {A functional central limit theorem for recursive residuals and applications in asymptotic statistics}, doi = {10.17904/ku.opus-774}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:824-opus4-7749}, school = {Katholische Universit{\"a}t Eichst{\"a}tt-Ingolstadt}, pages = {85 Seiten : Diagramme}, year = {2022}, abstract = {We study partial sums of recursive residuals, define a recursive residual partial sum process and prove that this process converges weakly against Brownian motion. For not normally distributed errors, the limit process has been known only for time series samples under assumptions that are difficult to verify, or in the case of triangular schemes of design points only under strong assumptions on the regression functions. For the first time, we determine the limit process for triangular schemes without the restrictive assumption of normally distributed errors and under very mild assumptions on the regression functions (they have to be left continuous, of bounded variation and linearly independent with respect to the L2 norm) that are of great benefit for practical applications. Crucial for the proof are Donsker's invariance principle for triangular schemes, a technique to factorize simultaneously a family of functions and Rubin's famous continuous mapping theorem. Our approach, based on Rubin's theorem, further allows us to compute the distribution of the limit process under local alternatives. With the help of these asymptotic results we are then able to define and study asymptotic tests and we give an example of an asymptotically uniformly most powerful test. We also give an introduction to the theory of weak convergence of finite measures, and present these classical results in a more general form than usual. We consider measures on perfectly normal spaces instead of metric spaces, and we consider filters of measures instead of sequences of measures.}, subject = {Lineare Regression}, language = {en} }