Limiting behaviour of a geometric-type estimator for tail indices

Let Z1, Z2,... be i.i.d. random variables with tail behaviour P (Z1> z) = r(z)e−Rz, where r is a regularly varying function at infinity and R is a positive constant. We consider the problem of estimating the exponential tail coef-ficient R, by methods mainly based on least squares considerations. Using a geometrical reasoning, we introduce a consistent estimator, whose values lay between the least squares estimates proposed by Schultze and Steinebach (1996). We investigate here the weak asymptotic properties of this geometric-type estimator, showing in particular that, under general conditions, its dis-tribution is asymptotically normal. The results are applied to the related problem of estimating the adjustment coefficient in risk theory (Csörgo ̋ and Steinebach (1991)) and a simulation study is performed in order to illustrate the finite sample behaviour of this estimator. 1 1 Key words and phrases. Adjustment coefficient, least squares estimators, order statistics, parameter estimation, regular variation, universal asymptotic normality