Smooth tail-index estimation

The two parametric distribution functions appearing in the extreme-value theory – the generalized extreme-value distribution and the generalized Pareto distribution – have log-concave densities if the extreme-value index γ∈[−1, 0]. Replacing the order statistics in tail-index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density ˆ f n leads to novel smooth quantile and tail-index estimators. These new estimators aim at estimating the tail index especially in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean-squared error.

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