Regularized Nonparametric Logistic Regression and Kernel Regularization 1

Regularization methods consist of a category of commonly used techniques to obtain robust solutions to ill-posed problems such as nonparametric regression and classification. In recent years, methods of regularization have also been successfully introduced to address some other classical problems in statistics, e.g. model/variable selection and dimension reduction. This thesis is composed of two major parts, both of which are within the framework of regularization methods. In the first part of this thesis, we are interested in the physics problem of detecting high energy signal neutrino events. We propose a modification to the traditional nonparametric penalized likelihood approach, to take into account the usage of importance sampling techniques in the generation of the training data from computer experiments. We try to estimate the multivariate logit function of the signal neutrino events in order to find the most powerful decision boundary at a certain significance level to optimally separate signal from background neutrinos. For simulated normal data, we compare this approach with a non-standard support vector machine (SVM) approach. The results suggest that in the case of weighted binary data, logistic regression is more appropriate than SVM in terms of finding individual level curves of the logit function. We also propose a diagnostic plot to check the goodness of fit of the result when