High-dimensional peaks-over-threshold inference

Fondeville, Raphaël de; Davison, A. C.

doi:10.1093/biomet/asy026

Cited by 73 publications

(95 citation statements)

References 47 publications

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“…The difficulty resides in the computation of high‐dimensional multivariate Gaussian distributions needed for the exponent function V and its partial derivatives

V_{τ_{i}}

. Unbiased Monte Carlo estimates of these quantities can be obtained, and Thibaud et al () and de Fondeville and Davison () suggest using crude approximations to reduce the computational time while maintaining accuracy; see also Genton et al (), who instead suggest using hierarchical matrix decompositions. Our method is not limited to these two models and could potentially be applied to any max‐stable model for which the functions V and

V_{τ_{i}}

are known and computable.…”

Section: Discussionmentioning

confidence: 99%

Full likelihood inference for max‐stable data

Huser

Dombry

Ribatet

et al. 2019

Stat

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We show how to perform full likelihood inference for max‐stable multivariate distributions or processes based on a stochastic expectation–maximization algorithm, which combines statistical and computational efficiency in high dimensions. The good performance of this methodology is demonstrated by simulation based on the popular logistic and Brown–Resnick models, and it is shown to provide computational time improvements with respect to a direct computation of the likelihood. Strategies to further reduce the computational burden are also discussed.

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“…The difficulty resides in the computation of high‐dimensional multivariate Gaussian distributions needed for the exponent function V and its partial derivatives

V_{τ_{i}}

V_{τ_{i}}

are known and computable.…”

Section: Discussionmentioning

confidence: 99%

Full likelihood inference for max‐stable data

Huser

Dombry

Ribatet

et al. 2019

Stat

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“…Thus, on average, ( c ∞ · C ) −1 simulations from the proposal distribution are needed to obtain an exact sample from the target distribution. Of course, to minimize the computational burden, for a given proposal density

{\overset{f}{true}}_{prop}

, the constant C should be chosen maximal subject to , that is,

C = \inf_{w \in {double-struckR}^{N}} \frac{{\tilde{f}}_{prop} (w)}{c_{\infty} f_{\max} (w)} .

Recently, de Fondeville and Davison () followed a similar idea and suggested to base the simulation of a general sup‐normalized spectral process

V {false(\cdot false)}^{\max} false/ false‖ V^{\max} ‖_{\infty}

on the relation

double-struckP ((), \frac{V^{\max}}{false‖ V^{\max} ‖_{\infty}} \in normald v) = \frac{false‖ \overset{V}{true} ‖_{\infty}}{double-struckE false‖ \overset{V}{true} ‖_{\infty}} double-struckP ((), \frac{\overset{V}{true}}{false‖ \overset{V}{true} ‖_{\infty}} \in normald v),

where

\overset{V}{true}

is a spectral process normalized with respect to another homogeneous functional r instead of the supremum norm, that is,

r false(\overset{V}{true} false) = 1

a.s. If

false‖ \overset{V}{true} ‖_{\infty}

is a.s. bounded from above by some constant, from the relation , we obtain an inequality of the same type as for the densities of

V {false(\cdot false)}^{\max} false/ false‖ V^{\max} ‖_{\infty}

and

\overset{V}{true} false(\cdot false) false/ false‖ \overset{V}{true} ‖_{}

…”

Section: Exact Simulation Via Rejection Samplingmentioning

confidence: 99%

“…For a Brown–Resnick process, it is well known that the sum‐normalized process

\overset{V}{true}

has the same distribution as

\exp false(W^{prop} false) false/ false‖ \exp false(W^{prop} false) ‖_{1}

where W prop has the density f prop from in Section with equal weights p 1 = … = p N = 1/ N (see also Dieker & Mikosch, ). Thus, in this case, the procedure proposed by de Fondeville and Davison () with r ( f ) = ‖ f ‖ 1 is equivalent to performing rejection sampling for

W^{\max}

with

{\overset{f}{true}}_{prop} = \sum_{i = 1}^{N} \frac{1}{N} f_{i}

as proposal distribution (Algorithm 2A). From (Equation ), it follows that rejection sampling can also be performed with

{\overset{f}{true}}_{prop} = \sum_{i = 1}^{N} p_{i} f_{i}

and arbitrary positive weights p 1 ,…, p N summing up to 1, because we have with

C = \min_{i = 1}^{N} p_{i}

.…”

Section: Exact Simulation Via Rejection Samplingmentioning

confidence: 99%

“…Section ); 2A.Algorithm 2 with proposal density

f_{prop} = \frac{1}{N} \sum_{i = 1}^{N} f_{i}

and C = 1/ N (equivalent to the procedure proposed in de Fondeville & Davison, , based on sum‐normalized spectral functions); 2B.Algorithm 2 with proposal density

{\overset{f}{true}}_{prop} = \sum_{i = 1}^{N} p_{i}^{*} false(ε^{*} false) g_{i, ε^{*}}

and C = C ( p ∗ , ε ∗ ) in where p ∗ ( ε ∗ ) ∈ [0,1] N and ε ∗ ∈ (0,1) are obtained as described in Section .…”

Section: Illustrationmentioning

confidence: 99%

“…In this paper, we will introduce alternative procedures for the simulation of

V^{\max}

or, equivalently,

W^{\max} = \log V^{\max}

, which are supposed to work for larger N , as well. To this end, we will modify a Markov chain Monte Carlo (MCMC) algorithm proposed by Oesting et al () and a rejection sampling approach based on the ideas of de Fondeville and Davison (). Both procedures have originally been designed to sample sup‐normalized spectral functions in general.…”

Section: Introductionmentioning

confidence: 99%

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Sampling sup‐normalized spectral functions for Brown–Resnick processes

Oesting

Schlather

Schillings

2019

Stat

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Sup‐normalized spectral functions form building blocks of max‐stable and Pareto processes and therefore play an important role in modelling spatial extremes. For one of the most popular examples, the Brown–Resnick process, simulation is not straightforward. In this paper, we generalize two approaches for simulation via Markov chain Monte Carlo methods and rejection sampling by introducing new classes of proposal densities. In both cases, we provide an optimal choice of the proposal density with respect to sampling efficiency. The performance of the procedures is demonstrated in an example.

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