Ana Ferreira

On optimising the estimation of high quantiles of a probability distribution

One of the major aims of one-dimensional extreme-value theory is to estimate quantiles outside the sample or at the boundary of the sample. The underlying idea of any method to do this is to estimate a quantile well inside the sample but near the boundary and then to shift it somehow to the right place. The choice of this “anchor quantile” plays a major role in the accuracy of the method. We present a bootstrap method to achieve the optimal choice of sample fraction in the estimation of either high quantile or endpoint estimation which extends earlier results by Hall and Weissman (1997) in the case of high quantile estimation. We give detailed results for the estimators used by Dekkers et al. (1989). An alternative way of attacking problems like this one is given in a paper by Drees and Kaufmann (1998).

The generalized Pareto process; with a view towards application and simulation

In extreme value statistics, the peaks-over-threshold method is widely used. The method is based on the generalized Pareto distribution characterizing probabilities of exceedances over high thresholds in

Representation of non-stationary time series of significant wave height with autoregressive models

\mathbb {R}^d

Extreme Values in Reliability

. We present a generalization of this concept in the space of continuous functions. We call this the generalized Pareto process. Differently from earlier papers, our definition is not based on a distribution function but on functional properties, and does not need a reference to a related max-stable process. As an application, we use the theory to simulate wind fields connected to disastrous storms on the basis of observed extreme but not disastrous storms. We also establish the peaks-over-threshold approach in function space.

Extreme Value Theory: An Introduction

A time series of significant wave height measured at the Frigg Platform, in the North Sea is modelled using Box-Jenkins autoregressive models. As this series is non-stationary two different transformations are performed and compared. Since this type of data has several gaps, the whole series was divided into subseries with no gaps which allows piecewise modelling of the initial set. Very similar statistical properties among these subsets were observed and very similar models were obtained. This allowed the derivation of one autoregressive model which would describe adequately the whole data set.

Extreme value theory

Extreme value (EV) theory provides probabilistic and statistical tools to deal with the largest or the smallest values of a sample, hence with tails of distribution functions. Applications of EV theory in reliability analysis can be found in many fields. For instance, in corrosion analysis, where pits of larger depth are of primary interest, as pitting corrosion can lead to the failure of metal structures, or, in strength of material problems where the principle of a chain being as strong as its weakest link dominates. In the following text, main concepts of EV theory such as the maximum domain of attraction condition and EV distributions, peaks over threshold and the generalized Pareto distributions are discussed. Estimation within these frameworks, along with applications, are exemplified. Keywords: minimum; maximum; Weibull; Frechet; Gumbel; Pareto; peaks over threshold (POT); extreme value index; return period; high quantile; failure probability; mean excess function