Comparing extreme models when the sign of the extreme value index is known

In the literature on analyzing extremes, both generalized Pareto distributions and Pareto distributions are employed to infer the tail of a distribution with a known positive extreme value index. Similar studies exist for a known negative extreme value index. Intuitively, one should not employ the generalized Pareto distribution in the case of knowing the sign of the extreme value index. In this work, we show that fitting a generalized Pareto distribution is equivalent to the model in Hall (1982) in the case of a negative extreme value index, in both improving the rate of convergence and including the bias term of the asymptotic results of that reference. When the extreme value index is known to be positive, we show that fitting a generalized Pareto distribution may be preferred in some cases determined by a so-called second-order parameter and the extreme value index itself.