Some Problems in Model Selection 1

This dissertation consists of three parts: the first two parts are related to smoothing spline ANOVA models; the third part concerns the Lasso and its related procedures in model selection. In Part I, by adopting the Cox proportional hazard model to quantify the hazard function, we propose a novel nonparametric model selection technique to analyze time to event data, within the framework of smoothing spline ANOVA models. Instead of the usual squared norms in the traditional smoothing spline ANOVA, our method employs a regularization with the penalty functional being the sum of the component norms. This method shrinks functional components and produces some components that are exactly zeros. It is an extension of the “COSSO” proposal by Lin and Zhang (2002) in Gaussian regression to hazard regression. To compute the estimate when the smoothing parameter is fixed, we develop an efficient algorithm based on a reformulation of the penalized partial likelihood. Approximations to the leave-out-one likelihood cross validation score are derived to choose the smoothing parameters. Both simulations and real examples suggest that our proposal is very powerful for model selection and component estimation. Part II of the thesis concerns penalized likelihood density estimation. We introduce a randomized Generalized Approximate Cross Validation score to