A class of distribution functions with less bias in extreme value estimation

Let X1,X2,… be i.i.d. random variables and let their distribution be in the domain of attraction of an extreme value distribution. Quite a few estimators of the extreme value index are known to be consistent under the domain of attraction conditions. When it comes to asymptotic normality a condition that is called second-order condition is very useful. The condition yields a speed of convergence of a polynomial rate. Then one gets asymptotically a normal distribution without bias, provided one restricts the number of tail observations used in the estimation to a certain polynomial of n, the total number of observations. We investigate what happens if the speed of convergence is faster than any polynomial rate. In that case one can use many more tail observations without creating bias.