Limiting behaviour of a geometric-type
estimator for tail indices
Abstract
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 coefficient 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
distribution is asymptotically normal. The results are applied to the related
problem of estimating the adjustment coefficient in risk theory (Csorgõ and
Steinebach (1991)) and a simulation study is performed in order to illustrate
the finite sample behaviour of this estimator.