- Numerical Contributions to the Asymptotic Theory of Robustness (2005)
- In the framework of this dissertation a software package – the R bundle RobASt – by means of the statistics software R has been developed. It includes all robust procedures introduced throughout the thesis. The dissertation itself consists of five parts and starts with a brief motivation, which makes precise why robust statistics is necessary. After that a detailed summary in German and English is given. Part I provides a description of the asymptotic theory of robustness (Chapter 1) which forms the basis of this thesis. It is based on Chapters 4 and 5 of Rieder (1994). Chapter 2 provides supplements to the asymptotic theory of robustness which have proved necessary for this thesis. More precisely, it contains results about: properties of the optimally robust influence curves (ICs), how one should proceed in an optimal way if the neighborhood radius is unknown – as mostly in practice, and the construction of estimates by means of the one-step method. At the end of Chapter 2 convergence of robust models is introduced which is related to the concept of convergence of experiments of Le Cam. Part II deals with optimally robust estimators for some non-standard models in robust statistics. These models are covered by the R package ROptEst which makes use of S4 classes and methods and is part of the R bundle RobASt. More precisely, the binomial (Chapter 3) and Poisson (Chapter 4) model, the exponential scale and Gumbel location model (Chapter 5) as well as the Gamma model (Chapter 6) are investigated. In particular, the binomial and Poisson model are used to study convergence of robust models. Using exponential scale and Gumbel location one can show that there is a connection between certain scale and location models via a log-transformation which also holds for the corresponding optimally robust ICs. Finally, the Gamma model is used to demonstrate how differentiable parameter transformations can be estimated in an optimally robust way. In Part III robust regression with random regressor and unknown error scale (Chapter 7) is treated where it is distinguished between simultaneous and separate estimation. In both cases the optimally robust estimators as well as robust estimators for several narrower classes of M estimators are considered. All these estimators are implemented in the R packages ROptRegTS and RobRex which are part of the R bundle RobASt. Numerical comparisons for several regressor distributions show that the various suboptimal M estimators may have very small but also huge efficiency losses. A further comparison of these and several other well-known robust estimators in case of normal location and scale is made in Chapter 8. These location and scale estimators are implemented in the R package RobLox which is part of the R bundle RobASt. In Part IV (Chapter 9) robust adaptivity in terms of two asymptotic MSE problems is defined. Hence, adaptivity is no longer only a dichotomous criterion but can be evaluated quantitatively in terms of efficiency loss. The various regression and time series models considered include models which are classically as well as robust-adaptive, models which are classically but not robust-adaptive, and finally models which are neither classically nor robust-adaptive. The numerical evaluations show that non-adaptivity depends in a crucial way on the considered model and may be very small in some models (e.g. AR(1) and MA(1)) but may be really huge in other models (e.g. ARCH(1)). Finally, in Part V (Chapter 10 – 12) asymptotic results are compared with their exact finite-sample counterparts. In case of a particular pseudo-loss function in terms of under-/overshoot probabilities an exact finite-sample as well as an asymptotic theory are available. As the analytic evaluation of the finite-sample risk turns out very difficult or even impossible for sample sizes larger than 2, algorithms based on the fast Fourier transform (FFT) have been developed to determine the exact finite-sample distribution of these differently robust estimators. Two interesting findings are: The (first order) asymptotics is too optimistic and the convergence towards the asymptotic values is better in case of total variation than in case of contamination neighborhoods. The appendix of this thesis contains supplementary results on the asymptotic theory of robustness for regression-type models (Appendix A), on the Kronecker product and the vec and vech operators (Appendix B) as well as on the convolution via FFT (Appendix C). Moreover, Appendix D provides a brief description of the R packages distrEx, RandVar, ROptEst and ROptRegTS which are part of the R bundle RobASt.