Max-domain of attraction for log-concave densities and smooth tail index estimation

Abstract

Both parametric distribution functions appearing in extreme value theory – the generalized extreme value distribution and the generalized Pareto distribution – have log–concave densities if the extreme value index $gamma$ lies in $[−1, 0]$. It is shown that all distribution functions $F$ having a log–concave density function belong to the max–domain of attraction of the generalized extreme value distribution. Given an i.i.d. sample $X_1, … , X_n$ where $X_i$ has a log–concave density $f$, the distribution function $F_n$ derived from the log–concave NPMLE $f_n$ is asymptotically equivalent to the empirical distribution function $F_n$. Replacing the order statistics in tail index estimators by the quantiles of $F_n$ leads to smoothed estimators of $gamma$. Monte Carlo simulations suggest that for finite $n$ these new estimators are highly accurate and well superior to their non–smoothed counterparts. If time permits, we discuss some problems in deriving asymptotical results.

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Palo Alto, United States