Max‐infinitely divisible models and inference for spatial extremes

Abstract

For many environmental processes, recent studies have shown that the dependence strength is decreasing when quantile levels increase. This implies that the popular max‐stable models are inadequate to capture the rate of joint tail decay, and to estimate joint extremal probabilities beyond observed levels. We here develop a more flexible modeling framework based on the class of max‐infinitely divisible processes, which extend max‐stable processes while retaining dependence properties that are natural for maxima. We propose two parametric constructions for max‐infinitely divisible models, which relax the max‐stability property but remain close to some popular max‐stable models obtained as special cases. The first model considers maxima over a finite, random number of independent observations, while the second model generalizes the spectral representation of max‐stable processes. Inference is performed using a pairwise likelihood. We illustrate the benefits of our new modeling framework on Dutch wind gust maxima calculated over different time units. Results strongly suggest that our proposed models outperform other natural models, such as the Student‐t copula process and its max‐stable limit, even for large block sizes.

Space-time extremes of severe US thunderstorm environments

“…In spatial extremes, the non-parametric bootstrap is often used for uncertainty assessment (Davison et al, 2013(Davison et al, , 2018Huser and Wadsworth, 2019), showing that researchers are aware of the shortcomings of using the estimated sandwich covariance matrix, but this cannot be said of using CLIC for model selection in a two-step setting. Many studies (e.g., Davison et al, 2013Davison et al, , 2018Huser and Genton, 2016;Huser et al, 2021) do not allow for the estimation of the margins.…”

“…We also contribute new methods for inference on max-stable fields. In environmental applications, data may exhibit asymptotic independence, in which case max-stable fields are unsuitable, and several subasymptotic models have been proposed to alleviate this (e.g., Huser and Wadsworth, 2019;Huser et al, 2021). One should always assess the validity of max-stable models in applications, and Gabda et al (2012) and Buhl and Klüppelberg (2016) proposed graphical diagnostics for data with standardized margins.…”

Exact simulation of continuous max-id processes

“…From a practical perspective one should note that it is sufficient to simulate from a finite PRM with intensity 1 {f (t)≥c} dµ(f ) for all t ∈ T and c > 0 to simulate the PRMs with intensities (9), since one can simply ignore those atoms which do not satisfy the constraints in (9).…”

“…Since μ is an infinite measure, (8) implies that the extremal point measure Ñ + (t i ) i ∈J 0 is finite. Therefore, we can use (9) to obtain a sample of Ñ + (t i ) i ∈J 0 via the simulation of finite PRMs. Combining the two simulated processes by taking pointwise maxima we obtain an approximation X of X, which satisfies Xt ∼ X t .…”

“…It is tempting to apply the same scheme to the simulation of a PRM with intensity 1 {f (t)≥c} dµ by drawing a random variable Z ∼ X t and, if Z ≥ c, drawing the extremal function according to Q t,Z . However, in contrast to the max-stable case, this procedure is only valid for the simulation of the extremal function at a single location t, since one cannot decide whether there would have been another extremal function at t = t, when the simulated extremal function at t does not satisfy the constraints in (9). Heuristically speaking, the reason for this lies in the inability to deduce a decomposition of the PRM with intensity 1 {f (t)>0} dµ into two independent parts, with the first part determining the magnitude of X t and the second part describing the corresponding normalized extremal function at t. This is a subtle, but crucial, difference.…”