Helmut Rieder

Robust asymptotic statistics

Contents: Preface.- Notation.- Chapter 1: Von Mises Functionals.- Chapter 2: Log Likelihoods.- Chapter 3: Asymptotic Statistics.- Chapter 4: Nonparametric Statistics.- Chapter 5: Optimal Influence Curves.- Chapter 6: Stable Constructions.- Chapter 7: Robust Regression.- Appendix A: Weak Convergence of Measures.- Appendix B: Some Functional Analysis.- Appendix C: Complements.- Bibliography.- Index.

The cost of not knowing the radius

Robust Statistics considers the quality of statistical decisions in the presence of deviations from the ideal model, where deviations are modelled by neighborhoods of a certain size about the ideal model. We introduce a new concept of optimality (radius-minimaxity) if this size or radius is not precisely known: for this notion, we determine the increase of the maximum risk over the minimax risk in the case that the optimally robust estimator for the false neighborhood radius is used. The maximum increase of the relative risk is minimized in the case that the radius is known only to belong to some interval [rl,ru]. We pursue this minmax approach for a number of ideal models and a variety of neighborhoods. Also, the effect of increasing parameter dimension is studied for these models. The minimax increase of relative risk in case the radius is completely unknown, compared with that of the most robust procedure, is 18.1% versus 57.1% and 50.5% versus 172.1% for one-dimensional location and scale, respectively, and less than 1/3 in other typical contamination models. In most models considered so far, the radius needs to be specified only up to a factor

A finite–sample minimax regression estimator

\rho\le \frac{1}{3}

A general robustness property of rank correlations

, in order to keep the increase of relative risk below 12.5%, provided that the radius–minimax robust estimator is employed. The least favorable radii leading to the radius–minimax estimators turn out small: 5–6% contamination, at sample size 100.

Contamination games in a robust k–sample model

The aim of the paper is to give a coherent account of the robustness approach based on shrinking neighborhoods in the case of i.i.d. observations, and add some theoretical complements. An important aspect of the approach is that it does not require any particular model structure but covers arbitrary parametric models if only smoothly parametrized. In the meantime, equal generality has been achieved by object-oriented implementation of the optimally robust estimators. Exponential families constitute the main examples in this article. Not pretending a complete data analysis, we evaluate the robust estimates on real datasets from literature by means of our R packages ROptEst and RobLox.

Robust Estimation of One Real Parameter When Nuisance Parameters are Present

For the model of simple linear regression through the origin and unconditional neighborhoods (errors–in–variables) as well as conditional neighborhoods (error–free–variables) of the L1 –type, an estimator of the slope of the regression line is derived which, among all estimators, is minimax at finite sample size and extends Hubers (1964) robust interval estimator of location

One-Sided Confidence About Functionals Over Tangent Cones

Motivated by the information bound for the asymptotic variance of M-estimates for scale, we define Fisher information of scale of any distribution function F on the real line as the supremum of all , where [phi] ranges over the continuously differentiable functions with derivative of compact support and where, by convention, 0/0:=0. In addition, we enforce equivariance by a scale factor. Fisher information of scale is weakly lower semicontinuous and convex. It is finite iff the usual assumptions on densities hold, under which Fisher information of scale is classically defined, and then both classical and our notions agree. Fisher information of finite scale is also equivalent to L2-differentiability and local asymptotic normality, respectively, of the scale model induced by F.

Robustness in Structured Models

Rank correlations are shown to be generally robust in the sense that the tests for independence, which they naturally define, have weakly equicontinuous error probabilities. The ordinary sample correlation coefficient is not robust in this respect.