Mathematical Sciences: Inference for Nonparametric Function Estimators

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A number of inference problems in nonparametric function estimation will be investigated during the course of this project. The problems to be considered include i) estimation of breakpoints for regression functions or their derivatives, ii) development of practical confidence bands for use in nonparametric regression, iii) construction of diagnostic goodness-of-fit tests using nonparametric function estimation methodology in a variety of settings, including censored data and additive models, and iv) spline smoothing with correlated errors. It is often reasonable to believe in many areas of science, such as biology, engineering, psychology, etc., that the data being collected is produced by two components: a nonrandom component, representing a characteristic of nature common to all individuals or subjects, and a random component that accounts for individual variation. This nonrandom component represents the understandable or predictable aspect of the phenomenon being studied and is therefore of considerable interest. Nonparametric function estimation is an area of statistics concerned with the development of flexible methods for estimating or describing the nonrandom components of data. The current research project deals with a variety of problems in nonparametric function estimation including i) the development of procedures for estimating a point where an abrupt change has occurred in some phenomenon (e.g., locating a point of sudden change in the discharge of water from a river or prices of a commodity) ii) the construction of accuracy measures (similar to those associated with popularity polls, etc.) to accompany nonparametric function estimators and iii) the development of statistical methods to assess the accuracy of models that have been postulated by scientists to describe various processes they are studying.