Given an ordered sample of either exceedances or upper order statistics which is to be modeled using a GPD with
distribution function \(F\), this function provides Falk's estimator of the shape parameter \(\gamma \in [-1,0]\)
if the endpoint
$$\omega(F) = \sup\{x \, : \, F(x) < 1\}$$
of \(F\) is known. Precisely,
$$\hat \gamma_{\rm{MVUE}} = \hat \gamma_{\rm{MVUE}}(k,n) = \frac{1}{k} \sum_{j=1}^k \log \Bigl(\frac{\omega(F)-H^{-1}((n-j+1)/n)}{\omega(F)-H^{-1}((n-k)/n)}\Bigr), \; \; k=2,\ldots,n-1$$
for \(H\) either the empirical or the distribution function based on the log--concave density estimator.
Note that for any \(k\), \(\hat \gamma_{\rm{MVUE}} : R^n \to (-\infty, 0)\). If \(\hat \gamma_{\rm{MVUE}}
\not \in [-1,0)\), then it is likely that the log-concavity assumption is violated.