Estimation Pareto tail index based on sample means

Abstract

We propose an estimator of the Pareto tail index m of a distribution, that competes well with the Hill, Pickands and moment estimators. Unlike the above estimators, that are based only on the extreme observations, the proposed estimator uses all observations; its idea rests in the tail behavior of the sample mean X¯n, having a simple structure under heavy-tailed F. The observations, partitioned into N independent samples of sizes n, lead to N sample means whose empirical distribution function is the main estimation tool. The estimator is strongly consistent and asymptotically normal as N → ∞, while n remains fixed. Its behavior is illustrated in a simulation study.

Keywords

• domain of attraction
• Pareto index
• strong embedding of empirical process
• tail behavior