165 research outputs found
Hidden tail chains and recurrence equations for dependence parameters associated with extremes of higher-order Markov chains
We derive some key extremal features for kth order Markov chains, which can
be used to understand how the process moves between an extreme state and the
body of the process. The chains are studied given that there is an exceedance
of a threshold, as the threshold tends to the upper endpoint of the
distribution. Unlike previous studies with k>1 we consider processes where
standard limit theory describes each extreme event as a single observation
without any information about the transition to and from the body of the
distribution. The extremal properties of the Markov chain at lags up to k are
determined by the kernel of the chain, through a joint initialisation
distribution, with the subsequent values determined by the conditional
independence structure through a transition behaviour. We study the extremal
properties of each of these elements under weak assumptions for broad classes
of extremal dependence structures. For chains with k>1, these transitions
involve novel functions of the k previous states, in comparison to just the
single value, when k=1. This leads to an increase in the complexity of
determining the form of this class of functions, their properties and the
method of their derivation in applications. We find that it is possible to find
an affine normalization, dependent on the threshold excess, such that
non-degenerate limiting behaviour of the process is assured for all lags. These
normalization functions have an attractive structure that has parallels to the
Yule-Walker equations. Furthermore, the limiting process is always linear in
the innovations. We illustrate the results with the study of kth order
stationary Markov chains based on widely studied families of copula dependence
structures.Comment: 35 page
Discussion of "Statistical Modeling of Spatial Extremes" by A. C. Davison, S. A. Padoan and M. Ribatet
Discussion of "Statistical Modeling of Spatial Extremes" by A. C. Davison, S.
A. Padoan and M. Ribatet [arXiv:1208.3378].Comment: Published in at http://dx.doi.org/10.1214/12-STS376B the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Accounting for choice of measurement scale in extreme value modeling
We investigate the effect that the choice of measurement scale has upon
inference and extrapolation in extreme value analysis. Separate analyses of
variables from a single process on scales which are linked by a nonlinear
transformation may lead to discrepant conclusions concerning the tail behavior
of the process. We propose the use of a Box--Cox power transformation
incorporated as part of the inference procedure to account parametrically for
the uncertainty surrounding the scale of extrapolation. This has the additional
feature of increasing the rate of convergence of the distribution tails to an
extreme value form in certain cases and thus reducing bias in the model
estimation. Inference without reparameterization is practicably infeasible, so
we explore a reparameterization which exploits the asymptotic theory of
normalizing constants required for nondegenerate limit distributions. Inference
is carried out in a Bayesian setting, an advantage of this being the
availability of posterior predictive return levels. The methodology is
illustrated on both simulated data and significant wave height data from the
North Sea.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS333 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Determining the Dependence Structure of Multivariate Extremes
In multivariate extreme value analysis, the nature of the extremal dependence
between variables should be considered when selecting appropriate statistical
models. Interest often lies with determining which subsets of variables can
take their largest values simultaneously, while the others are of smaller
order. Our approach to this problem exploits hidden regular variation
properties on a collection of non-standard cones and provides a new set of
indices that reveal aspects of the extremal dependence structure not available
through existing measures of dependence. We derive theoretical properties of
these indices, demonstrate their value through a series of examples, and
develop methods of inference that also estimate the proportion of extremal mass
associated with each cone. We apply the methods to UK river flows, estimating
the probabilities of different subsets of sites being large simultaneously
Bayesian spatial clustering of extremal behaviour for hydrological variables
To address the need for efficient inference for a range of hydrological extreme value problems, spatial pooling of information is the standard approach for marginal tail estimation. We propose the first extreme value spatial clustering methods which account for both the similarity of the marginal tails and the spatial dependence structure of the data to determine the appropriate level of pooling. Spatial dependence is incorporated in two ways: to determine the cluster selection and to account for dependence of the data over sites within a cluster when making the marginal inference. We introduce a statistical model for the pairwise extremal dependence which incorporates distance between sites, and accommodates our belief that sites within the same cluster tend to exhibit a higher degree of dependence than sites in different clusters. By combining the models for the marginal tails and the dependence structure, we obtain a composite likelihood for the joint spatial distribution. We use a Bayesian framework which learns about both the number of clusters and their spatial structure, and that enables the inference of site-specific marginal distributions of extremes to incorporate uncertainty in the clustering allocation. The approach is illustrated using simulations, the analysis of daily precipitation levels in Norway and daily river flow levels in the UK
A Poisson process reparameterisation for Bayesian inference for extremes
A common approach to modelling extreme values is to consider the excesses above a high threshold as realisations of a non-homogeneous Poisson process. While this method offers the advantage of modelling using threshold-invariant extreme value parameters, the dependence between these parameters makes estimation more dicult. We present a novel approach for Bayesian estimation of the Poisson process model parameters by reparameterising in terms of a tuning parameter m. This paper presents a method for choosing the optimal value of m that near-orthogonalises the parameters, which is achieved by minimising the correlation between the asymptotic posterior distribution of the parameters. This choice of m ensures more rapid convergence and ecient sampling from the joint posterior distribution using Markov Chain Monte Carlo methods. Samples from the parameterisation of interest are then obtained by a simple transform. Results are presented in the cases of identically and non-identically distributed models for extreme rainfall in Cumbria, UK
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