= Hill estimator
{c}
{title2=$\widehat\gamma_H=k^{-1}\sum_{j=1}^k\log(X_{(n-j+1)}/X_{(n-k)})$}
For $1\leq k<n$ and a positive threshold $t=X_{(n-k)}$, the <Hill estimator> averages the logarithmic ratios of the top $k$ <order statistics> to $t$. For a <Fréchet distribution> domain with shape $\alpha$, it estimates the reciprocal tail index $1/\alpha$. At an untied threshold it equals the <empirical distribution function> plug-in tail integral $\int_t^\infty(1-F_n(x))/(1-F_n(t))\,dx/x$. At a tied threshold the plug-in denominator counts strictly larger observations and the identity needs adjustment. Fixed $k$ does not by itself yield a consistent tail-index estimate.
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