A Monte Carlo simulation was used to generate data for a comparison of five robust regression estimation methods with ordinary least squares (OLS) under 36 different outlier data configurations. Two of the robust estimators, Least Absolute Value (LAV) estimation and MM estimation, are commercially available. Three authormodified variations on MM were also included (MM1, MM2, and MM3). Design parameters that were varied include sample size (n=60 and n=180), number of independent predictor variables (2, 3 and 6), outlier density (0%, 5% and 15%) and outlier location (2x,2y s, 8x8y s, 4x,8y s and 8x,4y s). Criteria on which the regression methods were measured are relative efficiency, bias and a test of the null hypothesis. Results indicated that MM2 was the best performing robust estimator on relative efficiency. The best performing estimator on bias was MM1. The best performing regression method on the test of the null hypothesis was MM2. Overall, the MM-type robust regression methods outperformed OLS and LAV on relative efficiency, bias, and the test of the null hypothesis.

by Man-Cheung Hau. / Bibliography: leaves 110-112 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1984

Thesis (M. Phil.)--University of Hong Kong, 1987.

published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Statistics / Master / Master of Philosophy

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.

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

In principal component analysis (PCA), the principal components (PC) are linear combinations of the variables that minimize some objective function. In the classical setup the objective function is the variance of the PC's. The variance of the PC's can be easily upset by outlying observations; hence, Chen and Li (1985) proposed a robust alternative for the PC's obtained by replacing the variance with an M-estimate of scale. This approach cannot achieve a high breakdown point (BP) and efficiency at the same time. To obtain both high BP and efficiency, we propose to use MM- and τ-estimates in place of the M-estimate. Although outliers may cause bias in both the direction and the size of the PC's, Chen and Li looked at the scale bias only, whereas we consider both. All proposed robust methods are based on the minimization of a non-convex objective function; hence, a good initial starting point is required. With this in mind, we propose an orthogonal version of the least median of squares (Rousseeuw and Leroy, 1987) and a new method that is orthogonal equivariant, robust and easy to compute. Extensive Monte Carlo study shows promising results for the proposed method. Orthogonal regression and detection of multivariate outliers are discussed as possible applications of PCA. / Science, Faculty of / Statistics, Department of / Graduate

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Estimation methods such as the Kalman filter identify best state estimates based on certain optimality criteria using a model of the system and the observations. A common assumption underlying the estimation is that the noise is Gaussian. In practical systems though, one quite frequently encounters thick-tailed, non-Gaussian noise. Statistically, contamination by this type of noise can be seen as inducing outliers among the data and leads to significant degradation in the KF. While many nonlinear methods to cope with non-Gaussian noise exist, a filter that is robust in the presence of outliers and maintains high statistical efficiency is desired. To solve this problem, a new robust Kalman filter framework is proposed that bounds the influence of observation, innovation, and structural outliers in a discrete linear system. This filter is designed to process the observations and predictions together, making it very effective in suppressing multiple outliers. In addition, it consists of a new prewhitening method that incorporates a robust multivariate estimator of location and covariance. Furthermore, the filter provides state estimates that are robust to outliers while maintaining a high statistical efficiency at the Gaussian distribution by applying a generalized maximum likelihood-type (GM) estimator. Finally, the filter incorporates the correct error covariance matrix that is derived using the GM-estimator's influence function. This dissertation also addresses robust state estimation for systems that follow a broad class of nonlinear models that possess two or more equilibrium points. Tracking state transitions from one equilibrium point to another rapidly and accurately in such models can be a difficult task, and a computationally simple solution is desirable. To that effect, a new robust extended Kalman filter is developed that exploits observational redundancy and the nonlinear weights of the GM-estimator to track the state transitions rapidly and accurately. Through simulations, the performances of the new filters are analyzed in terms of robustness to multiple outliers and estimation capabilities for the following applications: tracking autonomous systems, enhancing actual speech from cellular phones, and tracking climate transitions. Furthermore, the filters are compared with the state-of-the-art, i.e. the <i>H_â</i>-filter for tracking an autonomous vehicle and the extended Kalman filter for sensing climate transitions. / Ph. D.