Using a Bootstrap Method to choose the Sample Fraction in Tail Index Estimation

We use a subsample bootstrap method to get a consistent estimate of the asymptotically optimal choice of the sample fraction, in the sense of minimal mean squared error, which is needed for tail index estimation. Unlike previous methods our procedure is fully self contained. In particular, the method is not conditional on an initial consistent estimate of the tail index; and the ratio of the first and second order tail indices is left unrestricted, but we require the ratio to be strictly positive. Hence the current method yields a complete solution to tail index estimation as it is not predicated on a more or less arbitrary choice of the number of highest order statistics.