Clément Dombry

Bio: Clément Dombry is an academic researcher from University of Franche-Comté. The author has contributed to research in topics: Extreme value theory & Estimator. The author has an hindex of 18, co-authored 86 publications receiving 1085 citations. Previous affiliations of Clément Dombry include University of Poitiers & University of Burgundy.

Exact simulation of max-stable processes.

Abstract: Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult. Algorithms based on finite approximations are often inexact and computationally inefficient. We present a new algorithm for exact simulation of a max-stable process at a finite number of locations. It relies on the idea of simulating only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We further generalize the algorithm by Dieker & Mikosch (2015) for Brown-Resnick processes and use it for exact simulation via the spectral measure. We study the complexity of both algorithms, prove that our new approach via extremal functions is always more efficient, and provide closed-form expressions for their implementation that cover most popular models for max-stable processes and multivariate extreme value distributions. For simulation on dense grids, an adaptive design of the extremal function algorithm is proposed.

Conditional simulation of max-stable processes

Abstract: SUMMARY Since many environmental processes such as heat waves or precipitation are spatial in extent, it is likely that a single extreme event affects several loca tions and the areal modelling of extremes is therefore essential if the spatial dependence of e xtremes has to be appropriately taken into account. This paper proposes a framework for conditional simulations of max-stable processes and give closed forms for Brown‐Resnick and Schlather processes. We test the method on simulated data and give an application to extreme rainfall around Zurich and extreme temperature in Switzerland. Results show that the proposed framework provides accurate conditional simulations and can handle real-sized problems.

Functional regular variations, Pareto processes and peaks over threshold

Abstract: Although the last decades have seen many developments on max-stable processes, little is known on the limiting distribution of exceedances of stochastic processes. Paralleling the univariate extreme value theory, this work focuses on threshold exceedances of a stochastic process and their connections with regularly varying and generalized Pareto processes. More precisely we define an exceedance through a cost functional ` and show that the limiting (rescaled) distribution is a simple `–Pareto process whose spectral measure can be characterized. Several equivalent constructions for `–Pareto processes are given using either a constructive approach, either an homogeneity property or a peak over threshold stability. We also provide an estimator of the spectral measure and give some examples.

Conditional simulation of max-stable processes

Abstract: Since many environmental processes such as heat waves or precipitation are spatial in extent, it is likely that a single extreme event affects several locations and the areal modelling of extremes is therefore essential if the spatial dependence of extremes has to be appropriately taken into account. This paper proposes a framework for conditional simulations of max-stable processes and give closed forms for Brown-Resnick and Schlather processes. We test the method on simulated data and give an application to extreme rainfall around Zurich and extreme temperature in Switzerland. Results show that the proposed framework provides accurate conditional simulations and can handle real-sized problems.

Exact simulation of max-stable processes

Abstract: Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes simulation highly nontrivial. Algorithms based on finite approximations that are used in practice are often not exact and computationally inefficient. We will present two algorithms for exact simulation of a max-stable process at a finite number of locations. The first algorithm generalizes the approach by \citet{DM-2014} for Brown--Resnick processes and it is based on simulation from the spectral measure. The second algorithm relies on the idea to simulate only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We study the complexity of both algorithms and prove that the second procedure is always more efficient. Moreover, we provide closed expressions for their implementation that cover the most popular models for max-stable processes and extreme value copulas. For simulation on dense grids, an adaptive design of the second algorithm is proposed.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Applied Probability and Queues

Abstract: (1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

An Introduction To The Theory Of Point Processes

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