Lectures on Empirical Processes - Theory and Statistical Applications
This book is based on three series of lectures delivered at the Summer School on Empirical Processes organised by European Mathematical Society in September 2004. The titles of the lectures are: Empirical and quantile processes in the asymptotic theory of goodness-of-fit tests (Eustasio del Berrio), Topics on empirical processes (Paul Deheuvels) and Oracle inequalities (Sara van de Geer). Techniques developed through probability in a Banach space framework, strong approximations in the Hungarian style and tools from stochastic process theory appear in the book in connection with the main historical applications, as well as with others that are presently in development. The part written by del Barrio focuses on applications of empirical processes to the area of classical as well as recently developed goodness-of-fit tests. The second part, written by Deheuvels, concentrates on empirical distribution functions and empirical quantile functions and their functionals. In the last part, written by van de Geer, the construction of a statistical model is discussed. Attention is paid to penalized M-estimators and oracle inequalities. The book gives an excellent overview of the main techniques and results in the theory of empirical processes and its applications in statistics. It is assumed that the reader is familiar with probability theory and mathematical statistics. It can be used as a text of a course for PhD students.