| The single index model is an important tool in multivariate nonparametric regression. It generalizes linear regression, that is, it replaces the linear combination α0T x by a nonparametric component, η0(α 0Tx), where η 0(·) is an unknown univariate function. By reducing the dimensionality from multivariate predictors to a univariate index α 0Tx, this single index model avoids the so-called “curse of dimensionality” found in multivariate nonparametric regression. The penalized spline approach, on the other hand, has a number of advantages. As a direct fitting method, it is computationally expedient and flexible. This dissertation is concerned with penalized spline estimation for partially linear single index models, where the mean function has the form η0(α0 Tx) + β0 Tz.; In the first portion of the dissertation, we focus on the study of the model and the asymptotic properties of the estimators. The -consistency and asymptotic normality with a nontrivial smoothing parameter are rigorously shown for compact parameter spaces and then for general Euclidean spaces. These asymptotic results directly enable the further study of the joint inference for the parameters. In the remainder of the dissertation, we concentrate on a numerical study. We present real and simulated examples to illustrate the performance of penalized spline estimation for partially linear single index models. Our proposed estimation procedures are efficient. We further investigate the inference and the estimated asymptotic variance through a Monte Carlo variance study. Moreover, general Lq penalty functions are used. This allows added flexibility to the model when the mean function encounters discontinuity or spatial variation. |
