This project is the continuation of our research on the question of how intuitive and concise linear model concepts and techniques can be extended to nonparametric settings and on the development of nonparametric techniques for model diagnostics, dimensionality reduction, and assessing adequacy of particular classes of models. Some of the procedures considered in this research are based on conditional versions of the familiar regression, covariance, and correlation coefficients, where the conditioning is on covariates restricted to neighborhoods. The size of the neighborhood serves as a resolution scale, and dependence of the response on the covariates is measured and summarized at multiple scales. Other procedures are curve or surface estimators and estimators of integral functionals of distributions. Many of the estimators to be considered depend on smoothing parameters needed in estimation of curves and surfaces. A large part of the research addresses the problem of developing reliable data-based methods for smoothing parameter selection. Properties of estimators are studied using both asymptotic methods and Monte Carlo simulations.
Computer data bases of unprecedented size and complexity and the dramatic increase in computer power makes possible the development of more flexible models, concepts, and procedures, which can be used to study relationships between variables and to construct models without relying on rigid global assumptions. Much of the recent work in statistics has addressed this need for more general and flexible methods. This research further extends this work with a special focus on procedures that are counterparts of many commonly used linear model concepts and that expose important features in the data using intuitive and familiar ideas. The developed procedures are applied to financial, economic, insurance, medical, and other data.
