Partially smooth tail-index estimation for small samples

Small samples are a challenge in extreme value theory. Asymptotic results do not apply and many estimation techniques, e. g. maximum likelihood, are unstable. In such situations, imposing qualitative constraints on the empirical distribution function is known to greatly reduce variability. Distribution functions typically appearing in the extreme-value theory, e. g. the generalized extreme-value distribution or the generalized Pareto distribution, have monotone upper tails. Applying monotone density estimation to parts of initial kernel density estimators leads to partially smooth estimated distribution functions. Particularly in small samples, replacing the order statistics in tail-index estimators by their corresponding quantiles from partially smooth estimated distribution functions leads to improved tail-index estimators. Monte Carlo simulations demonstrate that the partially smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean-squared error.