Composite likelihood estimation for the Brown-Resnick process

Abstract

SUMMARY Genton et al. (2011) investigated the gain in efficiency when triplewise, rather than pairwise, likelihood is used to fit the popular Smith max-stable model for spatial extremes. We generalize their results to the Brown-Resnick model and show that the efficiency gain is substantial only for very smooth processes, which are generally unrealistic in applications.Some key words: Brown-Resnick process; Composite likelihood; Max-stable process; Pairwise likelihood; Smith process; Triplewise likelihood. INTRODUCTIONMax-stable processes are useful for the statistical modelling of spatial extreme events. No finite parametrization of such processes exists, but a spectral representation (de Haan, 1984) aids in constructing models. In a 1990 University of Surrey technical report, R. L. Smith proposed a max-stable model based on deterministic storm profiles which has become popular because it is simple, readily interpreted and easily simulated, but unfortunately it is too inflexible for realism in practice. Another popular model, the Brown-Resnick process, is based on intrinsically stationary log-Gaussian processes, can handle a wide range of dependence structures, and often provides a better fit to data; see, for example, or a 2012 University of North Carolina at Chapel Hill PhD thesis by Soyoung Jeon. Kabluchko et al. (2009) provided further underpinning for this process by showing that that under mild conditions, the BrownResnick process with variogram 2γ(h) = ( h /ρ) α (ρ > 0, 0 < α ≤ 2), where h is the spatial lag, is essentially the only isotropic limit of properly rescaled maxima of Gaussian processes. The Smith model is obtained by taking a Brown-Resnick process with variogram 2γ(h) = h T Σ −1 h for some covariance matrix Σ, corresponding after an affine transformation to taking α = 2, whereas found that 1/2 < α < 1 for the rainfall data they examined.Likelihood inference for max-stable models is difficult, since only the bivariate marginal density functions are known in most cases, and pairwise marginal likelihood is typically used (Padoan et al., 2010;Davison and Gholamrezaee, 2012;Huser and Davison, 2012). This raises the question whether some other approach to inference would be preferable. Genton et al. (2011) derived the general form of the likelihood function for the Smith model and showed that large efficiency gains can arise when fitting it using triplewise, rather than pairwise, likelihood. In this paper we extend their investigation to the Brown-Resnick process and show that for rougher models, more realistic than those considered by Genton et al. (2011), the efficiency gains are much less striking. Thus pairwise likelihood inference provides a good compromise between statistical and computational efficiency in many applications.

Geostatistics of Dependent and Asymptotically Independent Extremes

“…(7), and thus yields exact independence between maxima at such sites. Another possibility, which can be viewed as extending the Smith model (Huser and Davison 2013), is the Brown-Resnick process (Brown and Resnick 1977;Kabluchko et al 2009), sometimes called the geometric Gaussian process. This is constructed by using Eq.…”

“…When individual events are recorded, more efficient inference is feasible. Since the max-stable models are suitable only above some predetermined high threshold, inference is usually made using a censored approach (Huser andJeon and Smith 2012;Thibaud et al 2013). Furthermore, following Stephenson and Tawn (2005), Davison and Gholamrezaee (2012) and Wadsworth and Tawn (2013) show how to incorporate the occurrence times of extreme events, use of which both simplifies the likelihood and allows much more efficient inference in cases of moderate to low spatial dependence.…”

Efficient inference for spatial extreme value processes associated to log-Gaussian random functions

“…where φ d (·; ) denotes the d-dimensional multivariate normal probability density function with covariance matrix , composed of diagonal elements σ 2 i = σ 2 (s i ) and off-diagonal elements Huser & Davison (2013) use (7) to derive…”

“…The d-dimensional distribution functions for all such processes are known (Genton et al, 2011;Huser & Davison, 2013;Engelke et al, 2014), but owing to the exponential form of distribution function (4), even when the expectation is calculable, differentiation to yield high-dimensional densities produces an explosion of terms in the density: the d-variate density consists of B d summands, where B d is the dth Bell number. Furthermore, the known representations of the exponent function admit apparently awkward derivatives, which are themselves sums of several terms.…”

“…Furthermore, the known representations of the exponent function admit apparently awkward derivatives, which are themselves sums of several terms. Thus in practice tripletwise composite likelihood inference is possible, though Huser & Davison (2013) suggest that, except for the Gaussian extreme value process model, a special boundary case of the Brown-Resnick class, the relative gain in parameter estimation efficiency is not overwhelming.…”