The investigators propose to construct the robust Theil-Sen estimators (TSE's) in general regression models based on multivariate medians, to study their theoretical behaviors, and to explore their practical applications. The proposed TSE's are given for different regression models based on different multivariate medians. These models include multivariate linear regression, nonparametric regression, semiparametric regression, mixed and additive regression, penalized spline regression, local polynomial regression, wavelet-based smoothers, kriging, etc., while the multivariate medians include in particular those based on different depth functions such as the half space depth, the projection depth, the simplicial depth, the spatial depth, etc. The theory of depth functions can be viewed in part as a multivariate generalization of the univariate rank theory. Depths induce an ordering of all points from a center outward in a high dimensional space because of the lack of the linear ordering in the high dimension. The investigators specifically propose to: 1) Generalize depth functions of vector to matrix argument; 2) Investigate the uniqueness, robustness, consistency, and asymptotic normality; 3) Compare asymptotic relative efficiency of the proposed TSE's with other common estimators(e.g. least squares estimators), compare different multivariate-median-based TSE's, and compare with other robust estimators; 4) Calculate the complexities, implement algorithms and provide codes that can be accessed by other potential users; 5) Conduct statistical inference, perform simulations, and apply to real applications.
The Theil-Sen estimator is an estimator of the slope parameter in a simple linear regression model. It is robust to outliers, easy to compute, competitive to the least squares estimator, and has an intuitive geometric interpretation. Despite its many good properties, the TSE is vastly under-utilized because it is developed for a simple linear regression. In recent years semiparametric and nonparametric models have become a popular choice in statistical modeling. They now play an increasing important role in many areas of statistics since they are more realistic and flexible than parametric models. The proposed research will extend the robust TSE's to various useful semiparametric and non-parametric regression models, and will accordingly advance the theory of robust estimation in semiparametric models and provide more robust ways of analyzing data from many applications. Just like regression analysis which is so popularly used in almost every area of science, these proposed Theil-Sen estimators have wide applications, for example, in astronomy; in remote sensing; in geosciences(e.g. detecting trends of extreme rainfall series); in environmental sciences (e.g. trend analysis for ambient water quality such as detecting seasonal patterns, changes in rainfall, etc so as to assess the relationships among different factors; to help set water quality guidelines for impacted streams; etc.); in pattern recognitions (e.g. detection of road segments in noisy aerial images), in social sciences; and so forth. The proposed research will also involve training of graduate students for future researchers in statistics as well as providing selected undergraduate students with research experience.
