Tail index estimation depends for its accuracy on a precise choice of the sample fraction, i.e. the number of extreme order statistics on which the estimation is based. A complete solution to the sample fraction selection is given by means of a two step subsample bootstrap method. This method...

Estimators of the extreme-value index are based on a set of upper order statistics. We present an adaptive method to choose the number of order statistics involved in an optimal way, balancing variance and bias components. Recently this has been achieved for the similar but somewhat less...

Asymptotic tail probabilities for bivariate linear combinations of subexponential random variables are given. These results are applied to explain the joint movements of the stocks of reinsurers. Portfolio investment and retrocession practices in the reinsurance industry, for reasons of...

For samples of random variables with a regularly varying tail estimating the tail index has received much attention recently. For the proof of asymptotic normality of the tail index estimator second order regular variation is needed. In this paper we first supplement earlier results on...

We give a sufficient condition for i.i.d. random variables X1,X2 in order to have P{X1-X2>x} ~ P{|X1|>x} as x tends to infinity. A factorization property for subexponential distributions is used in the proof. In a subsequent paper the results will be applied to model fragility of financial markets.

We characterize second order regular variation of the tail sum of F together with a balance condition on the tails interms of the behaviour of the characteristic function near zero.

We prove that the probability distribution of Hill's estimator can be betterapproximated by a series of appropriate gamma distributions than by the limitingnormal distribution.

We prove that the probability distribution of Hill's estimator can be better approximated by a series of appropriate gamma distributions than by the limiting normal distribution.

The theory of stable probability distributions and their domains of attraction is derived in a direct way(avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varyingfunctions play an important role in the exposition.

Suppose are independent subexponential random variables with partial sums. We show that if the pairwise sums of the ’s are subexponential, then is subexponential and . The result is applied to give conditions under which as , where are constants such that is a.s. convergent. Asymptotic tail...