In this work we discuss tail index estimation for heavy-tailed distributions with an emphasis on robustness. After a short introduction we provide a theoretical background treating regular variation and its extensions. In particular, we consider second- and third-order properties of regularly varying functions and state uniform approximations. Based on this, classical results of Extreme Value Theory including limit distributions of normalized maxima and necessary and suffcient conditions for maximum domains of attraction (MDA) are discussed. In particular, we present a new two-parametric characterization of limiting distributions of normalized maxima naturally arising from second-order regular variation of the tail quantile function. Additionally we provide an interpretation of MDA-conditions for heavytailed distributions and derive their empirical counterparts. Generalized versions of empirical MDA-conditions lead to asymptotic expansions of the tail quantile process and the tail empirical process. Thereafter we discuss different tail index estimators, first considering an approach based on robustification of the Pareto-MLE. We then establish the parametric rate of convergence and quantify the robustness properties of the resulting Huberized Tail Index Estimator by the Inuence Function. Subsequently, classical tail index estimators based on relative excesses are considered. In particular these are linked to empirical versions of MDA-conditions. This relation also leads to some new classes of estimators, including p-Quantile Tail Index Estimators and Harmonic Moment Tail Index Estimators (HME). We derive the asymptotic properties of these classes and compare them with the well known Hill estimator. It turns out that the HME outperforms the Hill estimator in certain situations. Moreover, the parametric Huberized Tail Index Estimator shows a competitive behavior in comparison to the Hill estimator for small to moderate sample sizes. These asymptotic results are confirmed by simulations illustrating the finite sample behavior of corresponding estimators.We conclude by discussing the issue of tail index estimation for linear long memory processes with infinite second moments. A unifying characterization of long memory for strict stationary processes is proposed. Moreover, the tail index of a linear long memory process with α-stable innovations is estimated by a modified version of the Huberized Tail Index Estimator and asymptotic properties are derived
