Loi de Pareto

En théorie des probabilités, la loi de Pareto, d'après Vilfredo Pareto, est un type particulier de loi de puissance qui a des applications en sciences physiques et sociales. Elle permet notamment de donner une base théorique au « principe des 80-20 », aussi appelé principe de Pareto. Définition Soit la variable aléatoire X qui suit une loi de Pareto de paramètres (x,k), avec k un réel positif, alors la loi est caractérisée par : :\mathbb{P}(X>x)=\left(\frac{x_{\mathrm{m}}}{x}\right)^{k} \quad \textrm{ avec }\quad x \ge x_{\mathrm{m}} Densité de probabilité Les lois de Pareto sont des lois continues. La loi de Zipf, et son cas limite, la loi zêta, peuvent être considérées comme l'équivalent discret de la loi de Pareto. Il suit, de la définition donnée précédemment, que la densité de probabilité de X vérifie : f(x;k,x_{\mathrm{m}}) = k,\frac{x_{\mathrm{m}}^k}{x^{k+1}} pour x \ge x_{\mathrm{m}}. Le paramè

Contributions to Likelihood-Based Modelling of Extreme Values

Léo Raymond-Belzile

xtreme value analysis is concerned with the modelling of extreme events such as floods and heatwaves, which can have large impacts. Statistical modelling can be useful to better assess risks even if, due to scarcity of measurements, there is inherently very large residual uncertainty in any analysis. Driven by the increase in environmental databases, spatial modelling of extremes has expanded rapidly in the last decade. This thesis presents contributions to such analysis. The first chapter is about likelihood-based inference in the univariate setting and investigates the use of bias-correction and higher-order asymptotic methods for extremes, highlighting through examples and illustrations the unique challenge posed by data scarcity. We focus on parametric modelling of extreme values, which relies on limiting distributional results and for which, as a result, uncertainty quantification is complicated. We find that, in certain cases, small-sample asymptotic methods can give improved inference by reducing the error rate of confidence intervals. Two data illustrations, linked to assessment of the frequency of extreme rainfall episodes in Venezuela and the analysis of survival of supercentenarians, illustrate the methods developed. In the second chapter, we review the major methods for the analysis of spatial extremes models. We highlight the similarities and provide a thorough literature review along with novel simulation algorithms. The methods described therein are made available through a statistical software package. The last chapter focuses on estimation for a Bayesian hierarchical model derived from a multivariate generalized Pareto process. We review approaches for the estimation of censored components in models derived from (log)-elliptical distributions, paying particular attention to the estimation of a high-dimensional Gaussian distribution function via Monte Carlo methods. The impacts of model misspecification and of censoring are explored through extensive simulations and we conclude with a case study of rainfall extremes in Eastern Switzerland.

EPFL2019

Fractional extreme distributions

Lotfi Boudabsa, Thomas Simon

We consider three classes of linear differential equations on distribution functions, with a fractional order alpha is an element of [0; 1]. The integer case alpha = 1 corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent alpha-stable subordinator. From the analytical viewpoint, this distribution is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Frechet cases, and with a Le Roy function for the Gumbel case.

2020

Contributions to Likelihood-Based Modelling of Extreme Values

Léo Raymond-Belzile

EPFL2019

Contributions to Likelihood-Based Modelling of Extreme Values

Generalized Pareto Distribution for reliability of bridges exposed to fatigue

Fractional extreme distributions

Contributions to Likelihood-Based Modelling of Extreme Values

Generalized Pareto Distribution for reliability of bridges exposed to fatigue

Fractional extreme distributions