An estimator of the tail index based on increment ratio statistics

In this paper, we introduce an increment ratio statistic (IR N,m ) based estimator for estimation of the tail index of a heavy-tailed distribution. For i.i.d. observations depending on the zone of attraction of an α-stable law (0 < α < 2), the IR N,m statistic converges to a decreasing function L(α) as both the sample size N and bandwidth parameter m tend to infinity. We obtain a rate of decay of the bias EIR N,m -L(α) and mean square error E(IR N,m -L(α))2. A central limit theorem √N/m(IR N,m -EIR N,m) N(0,σ2(α)) is also obtained. Monte Carlo simulations show that our tail index estimator has quite good empirical mean square error and, unlike the Hill estimator, is not so sensitive to a change of bandwidth parameter m. © 2009 Springer Science+Business Media, Inc.