Given independent and identically distributed observations X(1), ..., X(n) from a Generalized Pareto distribution with shape parameter gamma in [-1,0], offers several estimates to compute estimates of gamma. The estimates are based on the principle of replacing the order statistics by quantiles of a distribution function based on a log--concave density function. This procedure is justified by the fact that the GPD density is log--concave for gamma in [-1,0].