High breakdown point, bounded influence and high efficiency at the Gaussian model are desired properties of robust regression estimators. Robustness of validity, robustness of efficiency and high breakdown point size and power are the fundamental goals in robust testing. The objective of this dissertation is to examine the finite-sample properties of robust estimators and tests, and to find some useful applications for them. This is accomplished by extensive Monte Carlo experiments and other inference techniques in various contamination situations. In the linear regression model with an outlying regressor and deviations from the normal error distribution, robust estimators demonstrate noticeable advantages over the standard LS and maximum likelihood (ML) estimators. Our findings reveal that the finite-sample behavior of the robust estimators is very different from their asymptotic properties. The robust properties of estimators carry over to test statistics based on these estimators. The robust tests we proposed can achieve to the large extent the fundamental goals in robust testing. Economic applications on modelling the household consumption behavior and testing for (G)ARCH effects show that one can capture big gains from the appropriate utilization of the robust methods even at very simple models.

This thesis focuses on practical methods for constructing robust nonlinear control systems. In general, the development of such control systems is characterized by the solution to one or more Hamilton-Jacobi partial differential equations (HJE). However, no general analytical solution has yet been obtained to solve this optimization problem. Solutions have thus far only been obtained under certain conditions. Therefore, the first significant contribution of this thesis is a method for obtaining analytical expressions for approximate solutions to a common form of HJE (under certain assumptions regarding the class of nonlinear systems used). Additionally, modern state space controller synthesis techniques typically result in state estimators of equal or greater dimension than the plant model. However, it is often desirable, or even necessary, to approximate these controllers by models of lower state dimension. Presently, methods for developing nonlinear state balancing transformations are not very well understood. Therefore, the second significant contribution of this thesis is a proper algorithm for the application of state balancing techniques to nonlinear control systems and the subsequent reduction of the number of control states. The method to be developed for state balancing is based on the above framework for constructing analytical solutions to the HJE. In this thesis we will make use of three existing robust nonlinear control methods from the literature. These three methods have the advantage that they can all be constructed from solutions to a single form of HJE. Thus, by developing a method for obtaining analytical expressions for the solution to a single form of HJE, we are able to develop explicit polynomial solutions for each of these three control methods. Due to the difficulties associated with quantifying robustness and performance properties for nonlinear systems, the effectiveness of the three control methods considered shall be demonstrated via numerical simulations. The particular applications of interest to us here are space systems. First, we will consider the attitude control of a single spacecraft. Second, we examine the problem of formation flying control for a pair of spacecraft. The third and final problem we consider is the control of a nonlinear mass-spring chain.

The use of a stochastic model to predict the likelihood of future outcomes forms an integral part of decision optimization under uncertainty. In classical stochastic modeling uncertain parameters are often assumed to be driven by a particular form of probability distribution. In practice however, the distributional form is often difficult to infer from the observed data, and the incorrect choice of distribution can lead to significant quality deterioration of resultant decisions and unexpected losses. In this thesis, we present new approaches for evaluating expected future performance that do not rely on an exact distributional specification and can be robust against the errors related to committing to a particular specification. The notion of comprehensive robustness is promoted, where various degrees of model misspecification are studied. This includes fundamental one such as unknown distributional form and more involved ones such as stochastic moments and moment outliers. The approaches are developed based on the techniques of moment-based optimization, where bounds on the expected performance are sought based solely on partial moment information. They can be integrated into decision optimization and generate decisions that are robust against model misspecification in a comprehensive manner. In the first part of the thesis, we extend the applicability of moment-based optimization to incorporate new objective functions such as convex risk measures and richer moment information such as higher-order multivariate moments. In the second part, new tractable optimization frameworks are developed that account for various forms of moment uncertainty in the context of decision analysis and optimization. Financial applications such as portfolio selection and option pricing are studied.

The use of a stochastic model to predict the likelihood of future outcomes forms an integral part of decision optimization under uncertainty. In classical stochastic modeling uncertain parameters are often assumed to be driven by a particular form of probability distribution. In practice however, the distributional form is often difficult to infer from the observed data, and the incorrect choice of distribution can lead to significant quality deterioration of resultant decisions and unexpected losses. In this thesis, we present new approaches for evaluating expected future performance that do not rely on an exact distributional specification and can be robust against the errors related to committing to a particular specification. The notion of comprehensive robustness is promoted, where various degrees of model misspecification are studied. This includes fundamental one such as unknown distributional form and more involved ones such as stochastic moments and moment outliers. The approaches are developed based on the techniques of moment-based optimization, where bounds on the expected performance are sought based solely on partial moment information. They can be integrated into decision optimization and generate decisions that are robust against model misspecification in a comprehensive manner. In the first part of the thesis, we extend the applicability of moment-based optimization to incorporate new objective functions such as convex risk measures and richer moment information such as higher-order multivariate moments. In the second part, new tractable optimization frameworks are developed that account for various forms of moment uncertainty in the context of decision analysis and optimization. Financial applications such as portfolio selection and option pricing are studied.

Thesis (M.S.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on April 15, 2008) Includes bibliographical references.

Thesis (Ph. D.)--UCLA, 2007. / Vita. Includes bibliographical references (leaves 93-96).

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to Department of Manufacturing Engineering and Engineering Management in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [169]-205)

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to Department of Management Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves 114-124)

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 178-183).

Robust estimators for multivariate location and dispersion should be ãn consistent and highly outlier resistant, but estimators that have been shown to have these properties are impractical to compute. The RMVN estimator is an easily computed outlier resistant robust ãn consistent estimator of multivariate location and dispersion, and the estimator is obtained by scaling the classical estimator applied to the gRMVN subseth that contains at least half of the cases. Several robust estimators will be presented, discussed and compared in detail. The applications for the RMVN estimator are numerous, and a simple method for performing robust principal component analysis (PCA), canonical correlation analysis (CCA) and factor analysis is to apply the classical method to the gRMVN subset.h Two approaches for robust PCA and CCA will be introduced and compared by simulation studies.