Mathematical Sciences: Applications of Smoothing Splines forInference and Data Analysis

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The first part of this research develops valid confidence sets for functionals of a spline estimate and the estimated smoothing parameter. Two approaches to confidence intervals are to treat the bias of the estimate as a random variable or to replace the bias by an upper bound. The first method yields short intervals that are accurate only in an average sense while the second method gives longer conservative intervals. Confidence intervals for the smoothing spline are derived from the distribution of the cross-validation function. The second part of this research applies spline methods to two ill-posed problems that arise in microscopy and soil science. Both of these topcs are related to solving an integral equation with discrete and noisy data. In this context, partial spline models can compensate for boundary effects while certain experimental designs may imporve the resolution of the estimate. This statistical research project addresses the general problem in data analysis of estimating a smooth function f, from functionals of f that are observed with error. Often a parametric form of f is unknown and it is important to consider nonparametric estimators of f. This project focuses on one nonparametric estimator that has been successful over a diverse range of applications, a cross-validated smoothing spline.