Mathematical Sciences: Some Problems in Nonparametric Regression

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Eubank proposes to examine a number of problems in nonparametric regression. One of the most interesting is the problem of robust spline smoothing. The typical spline estimator of a regression function is found by minimizing a sum of squared errors plus a penalty function. However the sum of squared errors is highly sensitive to outliers. By substituting absolute error for squared error (or another appropriate function of the error) one can make the estimator more robust to outliers. Eubanks will examine the effect of incorporating and estimating a scale factor for the error terms in these more appropriate and general functions of the errors. He will also examine jackknife confidence intervals for the underlying regression function. These nonparametric regression models will also be applied to the traditional analysis of variance, analysis of covariance and intra-class regression models. Another problem he will address is the incorporation of the covariance structure of order statistics in the estimation of quantile functions and their densities. The problem of relating two variables (such as height and weight) for pairs of observations on individuals is addressed by nonparametric regression. Finding a functional relationship between the two variables is made considerably harder if some information is miscoded or extremely unusual individuals are included in the data. Eubank will investigate methods of finding functional relationships that will be less sensitive to outliers in the sample and more representative of the whole population.