The P.O.T. method (Peaks Over Threshold) consists in using the generalized Pareto distribution (GPD) as an approximation for the distribution of excesses over a high threshold. In this work, we use a refinement of this approximation in order to estimate second order parameters of the model using the method of probability-weighted moments (PWM): in particular, this leads to the introduction of a new estimator for the second order parameter ρ, which will be compared to other recent estimators through some simulations. Asymptotic normality results are also proved. Our new estimator of ρ looks especially competitive when  |ρ|  is small.

Let F be a distribution function in the domain of attraction of an extreme-value distribution Hγ. If Fu is the distribution function of the excesses over u and Gγ the distribution function of the generalized Pareto distribution, then it is well known that Fu(x) converges to Gγ(x/σ(u)) as u tends to the end point of F, where σ is an appropriate normalizing function. We study the rate of (uniform) convergence to 0 of F̅u(x)-G̅γ((x+u-α(u))/σ(u)), where α and σ are two appropriate normalizing functions.

The P.O.T. (Peaks-Over-Threshold) approachconsists of using the Generalized ParetoDistribution (GPD)to approximate the distribution of excesses over a threshold.We use the probability-weighted momentsto estimate the parameters of the approximating distribution.We study the asymptotic behaviour ofthese estimators (in particular their asymptotic bias) and also thefunctional bias of the GPD as an estimate of thedistribution function of the excesses. We adapt penultimateapproximation results to the case where parameters are estimated.

Let F be a distribution function (d.f) in the domain of attraction of an extreme value distribution ${ H_{\gamma} }$; it is well-known that Fu(x), where Fu is the d.f of the excesses over u, converges, when u tends to s+(F), the end-point of F, to $G_{\gamma}(\frac{x}{\sigma(u)})$, where $G_{\gamma}$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function Λ which verifies $\lim_{u \rightarrow s_+(F)} \Lambda (u) =\gamma$ and is such that $\Delta(u)= \sup_{x \in[0,s_+(F)-u[} |\bar{F}_u(x) - \bar{G}_{\Lambda(u)} (x/ \sigma(u))| $converges to 0 faster than $d(u)=\sup_{x \in[0,s_+(F)-u[} |\bar{F}_u(x) - \bar{G}_{\gamma}(x/ \sigma(u))|$.