| Highly nonlinear relationships between a response variable and a set of regressor variables can be difficult to model across the entire regressor spaces. Parametric regression models can be biased by missing important anomalies associated with the nonlinear structure. On the other hand, nonparametric regression models may result in more variable estimates, especially for small sample sizes. Semi-parametric regression modeling produces a hybrid fit combining assets of both the parametric and nonparametric regression approaches. This dissertation will address issues and extensions of semi-parametric models for small sample design settings. A detailed case study motivating the utility of semi-parametric regression within aeronautics is provided. Critical to the success of semi-parametric regression is an effective method of choosing optimal smoothing parameters. Weighted cross-validation has been shown to be most effective in choosing optimal smoothing parameters for semi-parametric models. We demonstrate that the type of weighted cross-validation that is most appropriate largely depends upon the magnitude of the underlying variation relative to roughness/bumpiness of the true mean surface. Two data sets from the literature serve as the basis for extensive simulation studies which demonstrate the utility of our recommendations. Two other weighted cross-validation methods are developed for semi-parametric regression and guidance is offered as to the appropriateness of their use. |
