CRPS M-estimation for max-stable models

Max-stable processes provide canonical models for the dependence of multivariate extremes. Inference with such models has been challenging due to the lack of tractable likelihoods which has motivated use of composite likelihood methods Padoan et al. (J. Amer. Stat. Assoc. 105(489):263–277, 2010). In contrast, the finite dimensional cumulative distribution functions (CDFs) appear natural to work with, and are readily available or can be approximated well. Motivated by this fact, in this work we develop an M-estimation framework for max-stable models based on the continuous ranked probability score (CRPS) of multivariate CDFs. We start by establishing conditions for the consistency and asymptotic normality of the CRPS-based estimators in a general context. We then implement them in the max-stable setting and provide readily computable expressions for their asymptotic covariance matrices. The resulting point and asymptotic confidence interval estimates are illustrated over popular simulated models. They enjoy accurate coverages and offer an alternative to composite likelihood based methods.     

Storm processes and stochastic geometry

This paper is devoted to a prototype of max-stable models called the storm process. At first its spatial distribution is given in association with different observation supports. Then the compatibility relationships between extremal coefficients at various supports are completely characterized. Particular attention is paid to the special case where the storms are indicator functions of Poisson polytopes. Explicit formulae are found for the extremal coefficients with finite or convex supports. A new algorithm for exactly simulating the Poisson storm process in continuous space is also provided. Overall, the storm process can be used as a benchmark for comparing the performances of several estimators of extremal coefficients, or for model selection.     

Spectral representations of sum- and max-stable processes

To each max-stable process with α-Fréchet margins, α ∈ (0,2), a symmetric α-stable process can be associated in a natural way. Using this correspondence, we deduce known and new results on spectral representations of max-stable processes from their α-stable counterparts. We investigate the connection between the ergodic properties of a stationary max-stable process and the recurrence properties of the non-singular flow generating its spectral representation. In particular, we show that a stationary max-stable process is ergodic iff the flow generating its spectral representation has vanishing positive recurrent component. We prove that a stationary max-stable process is ergodic (mixing) iff the associated SαS process is ergodic (mixing). We construct non-singular flows generating the max-stable processes of Brown and Resnick.     

Stationary max-stable processes with the Markov property

We prove that the class of discrete time stationary max-stable process satisfying the Markov property is equal, up to time reversal, to the class of stationary max-autoregressive processes of order 1. A similar statement is also proved for continuous time processes.     

On the association of sum- and max-stable processes

We address the notion of association of sum- and max-stable processes from the perspective of linear and max-linear isometries. We establish the appealing result that these two classes of isometries can be identified on a proper space (the extended positive ratio space). This yields a natural way to associate to any max-stable process a sum-stable process. By using this association, we establish connections between structural and classification results for sum- and max-stable processes.     

High-Order Composite Likelihood Inference for Max-Stable Distributions and Processes

In multivariate or spatial extremes, inference for max-stable processes observed at a large collection of points is a very challenging problem and current approaches typically rely on less expensive composite likelihoods constructed from small subsets of data. In this work, we explore the limits of modern state-of-the-art computational facilities to perform full likelihood inference and to efficiently evaluate high-order composite likelihoods. With extensive simulations, we assess the loss of information of composite likelihood estimators with respect to a full likelihood approach for some widely used multivariate or spatial extreme models, we discuss how to choose composite likelihood truncation to improve the efficiency, and we also provide recommendations for practitioners. This article has supplementary material online.     

Maxima of moving maxima of continuous functions

Maxima of moving maxima of continuous functions (CM3) are max-stable processes aimed at modelling extremes of continuous phenomena over time. They are defined as Smith and Weissman’s M4 processes with continuous functions rather than vectors. After standardization of the margins of the observed process into unit-Fréchet, CM3 processes can model the remaining spatio-temporal dependence structure. CM3 processes have the property of joint regular variation. The spectral processes from this class admit particularly simple expressions given here. Furthermore, depending on the speed with which the parameter functions tend toward zero, CM3 processes fulfill the finite-cluster condition and the strong mixing condition. Processes enjoying these three properties also enjoy a simple expression for their extremal index. Next a method to fit CM3 processes to data is investigated. The first step is to estimate the length of the temporal dependence. Then, by selecting a suitable number of blocks of extremes of this length, clustering algorithms are used to estimate the total number of different profiles. The parameter functions themselves are estimated thanks to the output of the partitioning algorithms. The full procedure only requires one parameter which is the range of variation allowed among the different profiles. The dissimilarity between the original CM3 and the estimated version is evaluated by means of the Hausdorff distance between the graphs of the parameter functions.     

Spectral tail processes and max-stable approximations of multivariate regularly varying time series

A regularly varying time series as introduced in Basrak and Segers (2009) is a (multivariate) time series such that all finite dimensional distributions are multivariate regularly varying. The extremal behavior of such a process can then be described by the index of regular variation and the so-called spectral tail process, which is the limiting distribution of the rescaled process, given an extreme event at time 0. As shown in Basrak and Segers (2009), the stationarity of the underlying time series implies a certain structure of the spectral tail process, informally known as the “time change formula”. In this article, we show that on the other hand, every process which satisfies this property is in fact the spectral tail process of an underlying stationary max-stable process. The spectral tail process and the corresponding max-stable process then provide two complementary views on the extremal behavior of a multivariate regularly varying stationary time series.     

Linear sufficiency and linear admissibility in a continuous time Gauss–Markov model

This paper considers the problem of estimation in a linear model when a stochastic process instead of a random vector is observed. Estimators obtained as integrals of the observed process are studied. Characterizations of linear sufficiency and admissibility similar to those given in the classical linear model are obtained in this context. Moreover, a definition of generalized ridge estimators in continuous time is introduced and also a characterization of such estimators is given.     

Limit theory for the empirical extremogram of random fields

Regularly varying stochastic processes are able to model extremal dependence between process values at locations in random fields. We investigate the empirical extremogram as an estimator of dependence in the extremes. We provide conditions to ensure asymptotic normality of the empirical extremogram centred by a pre-asymptotic version. The proof relies on a CLT for exceedance variables. For max-stable processes with Fréchet margins we provide conditions such that the empirical extremogram centred by its true version is asymptotically normal. The results of this paper apply to a variety of spatial and space–time processes, and to time series models. We apply our results to max-moving average processes and Brown–Resnick processes.     

Conditional sampling for max-stable processes with a mixed moving maxima representation

This paper deals with the question of conditional sampling and prediction for the class of stationary max-stable processes which allow for a mixed moving maxima representation. We develop an exact procedure for conditional sampling using the Poisson point process structure of such processes. For explicit calculations we restrict ourselves to the one-dimensional case and use a finite number of shape functions satisfying some regularity conditions. For more general shape functions approximation techniques are presented. Our algorithm is applied to the Smith process and the Brown-Resnick process. Finally, we compare our computational results to other approaches. Here, the algorithm for Gaussian processes with transformed marginals turns out to be surprisingly competitive.     

On identification of the threshold diffusion processes

We consider the problems of parameter estimation for several models of threshold ergodic diffusion processes in the asymptotics of large samples. These models are the direct continuous time analogues of the well known in time series analysis threshold autoregressive models. In such models, the trend is switching when the observed process attaints some (unknown) values and the problem is to estimate it or to test some hypotheses concerning these values. The related statistical problems correspond to the singular estimation or testing, for example, the rate of convergence of estimators is T and not ?T{\sqrt{T}} as in regular estimation problems. We study the asymptotic behavior of the maximum likelihood and Bayesian estimators and discuss the possibility of the construction of the goodness-of-fit test for such models of observation.     

Asymptotic models and inference for extremes of spatio-temporal data

Recently there has been a lot of effort to model extremes of spatially dependent data. These efforts seem to be divided into two distinct groups: the study of max-stable processes, together with the development of statistical models within this framework; the use of more pragmatic, flexible models using Bayesian hierarchical models (BHM) and simulation based inference techniques. Each modeling strategy has its strong and weak points. While max-stable models capture the local behavior of spatial extremes correctly, hierarchical models based on the conditional independence assumption, lack the asymptotic arguments the max-stable models enjoy. On the other hand, they are very flexible in allowing the introduction of physical plausibility into the model. When the objective of the data analysis is to estimate return levels or kriging of extreme values in space, capturing the correct dependence structure between the extremes is crucial and max-stable processes are better suited for these purposes. However when the primary interest is to explain the sources of variation in extreme events Bayesian hierarchical modeling is a very flexible tool due to the ease with which random effects are incorporated in the model. In this paper we model a data set on Portuguese wildfires to show the flexibility of BHM in incorporating spatial dependencies acting at different resolutions.     

Spectral analysis of the covariance of the almost periodically correlated processes

This paper deals with the spectrum of the almost periodically correlated (APC) processes defined on . It is established that the covariance kernel of such a process admits a Fourier series decomposition, K(s+t, s) , whose coefficient functions b are the Fourier transforms of complex measures m, , which are absolutely continuous with respect to the measure m_o. Considering the APC strongly harmonizable processes, the spectral covariance of the process can be expressed in terms of these complex measures m.

The usual estimators for the second order situation can be modified to provide consistent estimators of the coefficient functions b from a sample of the process. Whenever the measures m are absolutely continuous with respect to the Lebesgue measure, so m(dλ)=f(λ) dλ, the estimation of the corresponding density functions f is considered. Under hypotheses on the covariance kernel K and on the coefficient functions b, we establish rates of convergence in quadratic mean and almost everywhere of these estimators.     

The contiguity of probability measures and asymptotic inference in continuous time stationary diffusions and Gaussian processes with known covariance

We establish contiguity of families of probability measures indexed by T, as T → ∞, for classes of continuous time stochastic processes which are either stationary diffusions or Gaussian processes with known covariance. In most cases, and in all the examples we consider in Section 4, the covariance is completely determined by observing the process continuously over any finite interval of time. Many important consequences pertaining to properties of tests and estimators, outlined in Section 5, will then apply.     

Approximation of optimal stopping problems

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a differential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.     

Wiener processes with random effects for degradation data

This article studies the maximum likelihood inference on a class of Wiener processes with random effects for degradation data. Degradation data are special case of functional data with monotone trend. The setting for degradation data is one on which n independent subjects, each with a Wiener process with random drift and diffusion parameters, are observed at possible different times. Unit-to-unit variability is incorporated into the model by these random effects. EM algorithm is used to obtain the maximum likelihood estimators of the unknown parameters. Asymptotic properties such as consistency and convergence rate are established. Bootstrap method is used for assessing the uncertainties of the estimators. Simulations are used to validate the method. The model is fitted to bridge beam data and corresponding goodness-of-fit tests are carried out. Failure time distributions in terms of degradation level passages are calculated and illustrated.     

Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on the notion of autoregressive Hilbert processes that represent a generalization of the classical autoregressive processes to random variables with values in a Hilbert space. A careful analysis reveals, in particular, that this approach is related to the theory of function estimation in linear ill-posed inverse problems. In the deterministic literature, such problems are usually solved by suitable regularization techniques. We describe some recent approaches from the deterministic literature that can be adapted to obtain fast and feasible predictions. For large sample sizes, however, these approaches are not computationally efficient.With this in mind, we propose three linear wavelet methods to efficiently address the aforementioned prediction problem. We present regularization techniques for the sample paths of the stochastic process and obtain consistency results of the resulting prediction estimators. We illustrate the performance of the proposed methods in finite sample situations by means of a real-life data example which concerns with the prediction of the entire annual cycle of climatological El Niño-Southern Oscillation time series 1 year ahead. We also compare the resulting predictions with those obtained by other methods available in the literature, in particular with a smoothing spline interpolation method and with a SARIMA model.     

An equivalent representation of the Brown-Resnick process

Brown and Resnick (1977) introduce a max-stable process that is obtained as a limit of maxima of independent Ornstein-Uhlenbeck processes. As shown in Kabluchko et al. (2009) this process is dissipative and it therefore admits a mixed moving maxima representation. We show that the distribution of the spectral functions in this representation equals a well-known diffusion, namely a standard Brownian motion with drift conditional on taking negative values only. This can be used for fast simulation methods.