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Doctoral Dissertation Announcement
Candidate: Themba Louis Nyirenda
Doctor of Philosophy
Title: Heteroscedastic Linear Model Estimation Based on Ranks: An Iterated Reweighted Least Squares Approach
Dr. Gerald L. Sievers, Chair
Dr. Joseph W. McKean
Dr. Magdalena Niewiadomska-Bugaj
Dr. Qiji Zhu
Date: Friday, May 22, 2009 9:00 a.m. - 11:00 a.m.
3302 Rood Hall
For standard estimators, data that are heteroscedastic in nature contain outlying values which can lead to poor performance. In this study, we present a robust iterative method for estimating the location and scale parameters in the general linear model, using a rank based method. It is assumed that the errors are symmetric about 0 and the variance function model is nonlinear with respect to the scale coefficients and the design. The function is known up to a scale constant.
We propose taking the logarithm of the absolute values of the variance function to linearize it. The rank estimation of the scale coefficients amounts to regressing logs of absolute residuals from an initial rank based fit on to the design. The resulting scale coefficient estimates are used to form scale constants in a weighted signed-rank method. Thus, iterating between these two rank based methods leads to the desired estimates that are obtained from linear model fits for both types of coefficients.
For the heteroscedastic linear model under consideration, this study has made the following contributions: (1) the asymptotic normality results that are established here show that the estimators are both consistent and highly (over) efficient; (2) in each estimation problem, the Iterated Reweighted Least Squares (IRWLS) formulation for rank methods of Sievers and Abebe (2004) is employed with the other parameter substituted by their corresponding estimate from an appropriate iteration; (3) the high efficiency and good robustness qualities of the proposed method are confirmed by simulation trials that were conducted in two-sample problem, several groups and general linear models; (4) the linear issue, a consequence of employing the log transformation, is also investigated and shown to be well curtailed by the proposed method; and (5) finally, the method is shown to outperform other methods when applied to real life data from a Psychiatric Clinical Trial containing two treatments, one covariate, and one confounding variable. Thus, for samples larger than 20, the method is highly robust and efficient under non-normal distributions.