Simultaneous tail index estimation

Abstract

The estimation of the extreme-value index γ based on a sample of independent and identically distributed random variables has received considerable attention in the extreme-value literature. However, the problem of combining data from several groups is hardly studied. In this paper we discuss the simultaneous estimation of tail indices when data on several independent data groups are available. The proposed methods are based on regression models linking tail related statistics to the extreme-value index and parameters describing the second order tail behaviour. For heavy-tailed distributions (γ >0), estimators are derived from an exponential regression model for rescaled log-spacings of successive order statistics as described in Beirlant et al. (1999) and Feuerverger and Hall (1999). Estimators for γ ∈ R are obtained using the linear model for UH-statistics given in Beirlant et al. (2000). In both cases, the optimal number of extremes to be used in the estimation is derived from the asymptotic mean squared error matrix.