TY - THES
T1 - Estimation of a regression function by maxima of minima of linear functions
A1 - Clausen,Conny
Y1 - 2011/04/01
N2 - The estimation of a multivariate regression function from independent and identically distributed random variables is considered. First we propose and analyse estimates which are defined by minimisation of the empirical L_{2} risk over a class of functions consisting of maxima of minima of linear functions. It is shown that the estimates are strongly universally consistent. Moreover results concerning the rate of convergence of the estimates with data-dependent parameter choice using "splitting the sample'; are derived in the case of an unbounded response variable. In particular it is shown that, for smooth regression functions satisfying the assumptions of single index models, the estimate is able to achieve (up to some logarithmic factor) the corresponding optimal one—dimesional rate of convergence. In this context it is remarkable that this newly proposed estimate can be computed in applications (see the appendix). Furthermore an L_{2} boosting algorithm for estimation of a regression function is presented. This method repeatedly fits a function from a fixed function space to the residuals of the data and the number of iteration steps is chosen data—dependently by "splitting the sample';. A general result concerning the rate of convergence of the algorithm is derived in the case of an unbounded response variable. Finally this method is used to fit a sum of maxima of minima of linear functions to a given set of data. The derived rate of convergence of the corresponding estimate does not depend on the dimension of the observation variable.
KW - Regressionsfunktion
KW - Funktionenraum
CY - Saarbrücken
PB - Saarländische Universitäts- und Landesbibliothek
AD - Postfach 151141, 66041 Saarbrücken
UR - http://scidok.sulb.uni-saarland.de/volltexte/2011/3520
ER -