Tail index estimation with a fixed tuning parameter fraction

Abstract

Semi-parametric tail index estimators, such as the Hill, Harmonic Moment, Pickands, and Dekkers, Einmahl and de Haan estimators, rely upon a tuning parameter that typically grows with sample size n. Proper selection of this tuning parameter k=k(n) is crucial for good practical performance, although asymptotic theory dictates that 1/k+k/n→0 as n→∞. A similar issue presents itself in the bandwidth literature in spectral density estimation and recent research shows that the use of asymptotic distributions when the bandwidth is a fixed ratio of sample size yields improved approximations to finite-sample distributions. Here, we study some semi-parametric tail index estimators utilizing the same perspective where k=bn and b∈(0,1) is a fixed constant. This allows us to derive asymptotic bias and variance expressions which are compatible with the small-b conventional theory. Our simulations corroborate that the finite-sample bias and variance are well described by the asymptotic bias and variance quantities arising from our fixed bandwidth ratio theory.