A sequence of equivariant estimators of a location parameter, which is asymptotically most robust with respect to bias oscillation function, is derived for two types of disturbances: e-contamination and Kolmogorov-Levy neighbourhoods. The sequence consists of properly chosen order statistics modified by adding a constant. As examples, the most bias-robust estimators for unimodal symmetric, Weibull, double-exponential and beta distributions are presented.
Asymptotic robustness of estimators of scale parameter with respect to scale invariant bias oscillation function is studied for two types of disturbances. In the case of £-contamination, the most robust sequence of equivariant estimators for model distribution with a positive support and the most robust sequence of equivariant symmetric estimators for symmetric model distribution are constructed. In the case of Kolmogorov-Levy neighbourhoods, the solution is derived without any assumptions about the model distribution. As examples, the most bias-robust estimators for uniform, Pareto, Weibull, Laplace, normal, Cauchy and double-exponential distributions are presented.
In gaussian linear models with known matrices covariance, the problem of robust estimation of a given linear function f of variance components is considered. An estimator of robust is constructed which is the most stable (most model-robust) to changes of the kurtosis of the original distributions.
Standard statistical procedures for variance in Gaussian models are not robust against departures from normality. One of the possible reasons is that the variance of the variance estimate depens on kurtosis of the underlying distribution. In the paper, the most robust estimate of the variance in a class of quadratic forms is constructed.
For the exponential-scale and Pareto-shape testing problems the unbiased tests based on order statistics are considered. Under the violations defined by hazards and of the contamination type the tests with the most robust (stable) power functions axe explicitly constructed.
The problem of measuring the Bayesian robustness is considered. An upper bound for the oscillation of a posterior functional in terms of the Kolmogorov distance between the prior distributions is given. The norm of the Frechet derivative as a measure of local sensitivity is presented. The problem of finding optimal statistical procedures is presented.
In the paper, a numerical analysis of robustness of Student f-test, Wilcoxon-Mann-Whitney test and a sign test to some dependencies of observations is presented. A Monte Carlo approach has been applied.