Mathematical Sciences: Topics in Nonparametric Function Estimation

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Professor Paul L. Speckman will investigate several statistical problems in nonparametric curve and surface estimation, the problem of recovering smooth curves and surfaces from noisy data. There are a number of general methods for curve estimation which have been extensively studied including kernel estimators and smoothing splines. This project is aimed at applications of these methods in two specific areas, constructing confidence sets for curves and estimating change-points (or breakpoints) in otherwise smooth curves. The first part of the project is aimed at developing reliable, practical confidence bands for unknown smooth curves. The investigator will derive the necessary asymptotic theory to construct confidence bands based on cross-validated or otherwise data-driven smoothers, and he will study their finite sample properties by computer simulation. The second part of the project will employ related distibutional theory to develop methods for detection and estimation of change-points and estimation of the magnitude of the change. In this context, change-points are defined to be points where jump discontinuities in an otherwise smooth response function or its derivatives occur. These change-point models are useful in a variety of applications including longitudinal studies of patients in certain kinds of clinical trials. Professor Paul L. Speckman will investigate several statistical problems in an area known as nonparametric curve and surface estimation, the problem of recovering smooth curves and surfaces from noisy data. Such problems arise in a wide variety of areas of science such as economics, atmospheric sciences, engineering and medicine. There are a number of methods currently available for estimating such curves. The first problem to be addressed by Professor Speckman will be that of giving realistic and practical error bounds for these estimates. This will enable investigators using nonparametric curve estimators to know how good their results are. Theoretical results will be obtained and their applicability will be studied through computer simulation. The second problem to be studied, known as a change-point problem, pertains to situations where there is a sudden change in an otherwise smooth process. One example is an industrial process when there is a sudden shift in the quality of the output. Other examples come from the study of certain diseases in humans. One specific case in which the new methods will be tested is a certain form of muscular dystrophy where muscle strength abruptly begins to decline at the onset of the disease. Techniques to be developed in this project will enable researchers to detect when such change-points have occurred and to estimate the size of the change.