| Properties of robust statistics have been extensively studied in the univariate setting when the underlying model is presumed to be symmetric, and in the multivariate case when the underlying model is presumed to be elliptically symmetric. Much less attention has been given to the behavior of robust statistics under asymmetric models. The goal of this dissertation is thus to obtain theoretical results for robust statistics under asymmetric models. To this end, local asymmetric alternatives to symmetric and elliptically symmetric distributions are considered. A key tool used in obtaining the theories presented in this dissertation is the LeCam's lemmas on contiguity.;The classes of robust univariate statistic considered here are the M-estimates, onestep version of the M-estimates, the W-estimates and the trimmed means. The classes of robust multivariate statistics considered are the M-estimates, the S-estimates, the CM-estimates and the MM-estimates, which are all treated under the unified framework of M-estimates with auxiliary scale, as well as their one-step versions. Asymptotic distributions of these statistics are obtained under local mixture models and skew-symmetric models. The asymptotic properties for the MM-estimates, even under elliptical symmetry, are the first such results for the multivariate MM-estimates.;Under asymmetry, different robust statistics for location are not consistent with each other, i.e. they are estimating different notions of central tendency. Likewise, in the multivariate setting, under non-elliptical distributions, the different scatter statistics are again not consistent with each other and are reflecting different structures of the underlying distribution. This suggests the difference in location statistics can be used to detect asymmetry and the comparison of different scatter statistics can be used to detect deviations from elliptical symmetry.;Consequently, new classes of tests for symmetry and for elliptical symmetry are introduced in this dissertation based upon the comparisons of different location statistics and different scatter statistics respectively. Furthermore, the asymptotic null distributions of the proposed test statistics are derived as well as their local power functions under contiguous mixture distributions. The local power functions help provide some guidelines for choosing the proper tuning constant of the proposed tests. |
