Mathematical Sciences: Topics in Nonparametric Analysis and Model Building

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It is proposed to address the question of how intuitive and concise linear model concepts and techniques can be extended to nonparametric settings and it is also proposed to develop nonparametric techniques for model diagnostics that can be used for dimensionality reduction and to address the question of adequacy of particular models. More precisely, it is proposed to focus on procedures that are counterparts to such commonly used linear model ideas as regression coefficients, correlation coefficients, ANOVA decompositions, and principal components. In addition, it is proposed to consider nonparametric tools for model building, identification, and diagnostics including tests for linearity, partial linearity, additivity, etc. All estimators to be considered depend on smoothing parameters needed in estimation of curves and surfaces. A large part of the research will address the problem of developing reliable data-based methods for smoothing parameter selection. With the advent of computer data bases of unprecedented size and complexity and with dramatic increase in computer power, it has become increasingly more desirable and possible to develop general models, concepts, and procedures that can be used to study relationships between variables and to construct models less dependent on specific assumptions. It is proposed to extend commonly used linear model ideas and techniques to this more general setting and to assess the adequacy of models with simpler structure.