Multivariate extreme value distributions

Max-stable processes are the natural analogues of the generalized extreme-value distribution when modelling extreme events in space and time. Under suitable conditions, these processes are asymptotically justified models for maxima of independent replications of random fields, and they are also suitable for the modelling of extreme measurements over high thresholds. This paper shows how a pairwise censored likelihood can be used for consistent estimation of the extremes of space-time data under mild mixing conditions, and illustrates this by fitting an extension of a model of Schlather (2002) to hourly rainfall data. A block bootstrap procedure is used for uncertainty assessment. Estimator efficiency is considered and the choice of pairs to be included in the pairwise likelihood is discussed. The proposed model fits the data better than some natural competitors.

Space–time modelling of extreme events

https://kellogg.northwestern.edu/faculty/todorov/htm/papers/jtd.pdf

Jump tails, extreme dependencies, and the distribution of stock returns

We propose an exact simulation method for Brown-Resnick random fields, building on new representations for these stationary max-stable fields. The main idea is to apply suitable changes of measure.

Exact simulation of Brown-Resnick random fields at a finite number of locations

Max-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several representations of max-stable random fields have been proposed in the literature. One such representation is based on a limit of normalized and scaled pointwise maxima of stationary Gaussian processes that was first introduced by Kabluchko, Schlather and de Haan (2009). 
This paper deals with statistical inference for max-stable space-time processes that are defined in an analogous fashion. We describe pairwise likelihood estimation, where the pairwise density of the process is used to estimate the model parameters and prove strong consistency and asymptotic normality of the parameter estimates for an increasing space-time dimension, i.e., as the joint number of spatial locations and time points tends to infinity. A simulation study shows that the proposed method works well for these models.

Statistical inference for max-stable processes in space and time

Max-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several families of max-stable random fields have been proposed in the literature. One such representation is based on a limit of normalized and rescaled pointwise maxima of stationary Gaussian processes that was first introduced by Kabluchko and co-workers. This paper deals with statistical inference for max-stable space–time processes that are defined in an analogous fashion. We describe pairwise likelihood estimation, where the pairwise density of the process is used to estimate the model parameters. For regular grid observations we prove strong consistency and asymptotic normality of the parameter estimates as the joint number of spatial locations and time points tends to ∞. Furthermore, we discuss extensions to irregularly spaced locations. A simulation study shows that the method proposed works well for these models.

Max-stable processes have proved to be useful for the statistical modeling of spatial extremes. For statistical inference it is often assumed that there is no temporal dependence; i.e., that the observations at spatial locations are independent in time. In a first approach we construct max-stable space–time processes as limits of rescaled pointwise maxima of independent Gaussian processes, where the space–time covariance functions satisfy weak regularity conditions. This leads to so-called Brown–Resnick processes. In a second approach, we extend Smith’s storm profile model to a space–time setting. We provide explicit expressions for the bivariate distribution functions, which are equal under appropriate choice of the parameters. We also show how the space–time covariance function of the underlying Gaussian process can be interpreted in terms of the tail dependence function in the limiting max-stable space–time process.

Max-stable processes for modeling extremes observed in space and time

In this article we analyze the extremal behavior of wind speed with a measurement frequency of 8 Hz, measured on three meteorological masts in Denmark. In the first part of this article we set up a conditional model for the time series consisting of threshold exceedances from maxima per second for two consecutive days. The model directly captures the nonstationary nature of wind speed during the day. Conditional on previous wind speed values with recorded exceedance, we assume a Markov-like structure for exceedances, where the conditional distribution follows the generalized Pareto distribution. In addition, we analyze the dependence structure in large wind speed values between different masts by using bivariate extreme value models. The initial motivation for this research was in the context of renewable energy. Specifically, the extremal dynamics of wind speed at small time scales plays a critical role for designing and locating turbines on wind farms.

Extreme Value Analysis of Multivariate High Frequency Wind Speed Data

Regularly varying space-time processes have proved useful to study extremal dependence in space-time data. We propose a semiparametric estimation procedure based on a closed form expression of the extremogram to estimate parametric models of extremal dependence functions. We establish the asymptotic properties of the resulting parameter estimates and propose subsampling procedures to obtain asymptotically correct confidence intervals. A simulation study shows that the proposed procedure works well for moderate sample sizes and is robust to small departures from the underlying model. Finally, we apply this estimation procedure to fitting a max-stable process to radar rainfall measurements in a region in Florida. Complementary results and some proofs of key results are presented together with the simulation study in the supplement [Buhl et al. (2018)].

/pdf/semiparametric-estimation-for-isotropic-max-stable-space-1bc35svond.pdf

Christina Steinkohl

Papers

Statistical inference for max-stable processes in space and time

Statistical inference for max-stable processes in space and time

Max-stable processes for modeling extremes observed in space and time

Extreme Value Analysis of Multivariate High Frequency Wind Speed Data

Semiparametric estimation for isotropic max-stable space-time processes