arXiv subjects

Jonathan A. Tawn

Publications and source records attributed to Jonathan A. Tawn.

24 records · Page 2Linked to original sources

arXiv2016

Extreme Events of Markov Chains

The extremal behaviour of a Markov chain is typically characterized by its tail chain. For asymptotically dependent Markov chains existing formulations fail to capture the full evolution of the extreme event when the chain moves out of the extreme tail region and for asymptotically independent chains recent results fail to cover well-known asymptotically independent processes such as Markov processes with a Gaussian copula between consecutive values. We use more sophisticated limiting mechanisms that cover a broader class of asymptotically independent processes than current methods, including an extension of the canonical Heffernan-Tawn normalization scheme, and reveal features which existing methods reduce to a degenerate form associated with non-extreme states.

math.PR↗

arXiv2016

Bayesian Uncertainty Management in Temporal Dependence of Extremes

Both marginal and dependence features must be described when modelling the extremes of a stationary time series. There are standard approaches to marginal modelling, but long- and short-range dependence of extremes may both appear. In applications, an assumption of long-range independence often seems reasonable, but short-range dependence, i.e., the clustering of extremes, needs attention. The extremal index $0<θ\le 1$ is a natural limiting measure of clustering, but for wide classes of dependent processes, including all stationary Gaussian processes, it cannot distinguish dependent processes from independent processes with $θ=1$. Eastoe and Tawn (2012) exploit methods from multivariate extremes to treat the subasymptotic extremal dependence structure of stationary time series, covering both $0<θ<1$ and $θ=1$, through the introduction of a threshold-based extremal index. Inference for their dependence models uses an inefficient stepwise procedure that has various weaknesses and has no reliable assessment of uncertainty. We overcome these issues using a Bayesian semiparametric approach. Simulations and the analysis of a UK daily river flow time series show that the new approach provides improved efficiency for estimating properties of functionals of clusters.

stat.ME↗

arXiv2015

Conditioned limit laws for inverted max-stable processes

Max-stable processes are widely used to model spatial extremes. These processes exhibit asymptotic dependence meaning that the large values of the process can occur simultaneously over space. Recently, inverted max-stable processes have been proposed as an important new class for spatial extremes which are in the domain of attraction of a spatially independent max-stable process but instead they cover the broad class of asymptotic independence. To study the extreme values of such processes we use the conditioned approach to multivariate extremes that characterises the limiting distribution of appropriately normalised random vectors given that at least one of their components is large. The current statistical methods for the conditioned approach are based on a canonical parametric family of location and scale norming functions. We study broad classes of inverted max-stable processes containing processes linked to the widely studied max-stable models of Brown-Resnick, Schlather and Smith, and identify conditions for the normalisations to either belong to the canonical family or not. Despite such differences at an asymptotic level, we show that at practical levels, the canonical model can approximate well the true conditional distributions.

math.PR↗

arXiv2014

Stochastic Ordering under Conditional Modelling of Extreme Values: Drug-Induced Liver Injury

Drug-induced liver injury (DILI) is a major public health issue and of serious concern for the pharmaceutical industry. Early detection of signs of a drug's potential for DILI is vital for pharmaceutical companies' evaluation of new drugs. A combination of extreme values of liver specific variables indicate potential DILI (Hy's Law). We estimate the probability of severe DILI using the Heffernan and Tawn (2004) conditional dependence model which arises naturally in applications where a multidimensional random variable is extreme in at least one component. We extend the current model by including the assumption of stochastically ordered survival curves for different doses in a Phase 3 study.

stat.AP↗

arXiv2012

Dependence Properties of Multivariate Max-Stable Distributions

For an m-dimensional multivariate extreme value distribution there exist 2^{m}-1 exponent measures which are linked and completely characterise the dependence of the distribution and all of its lower dimensional margins. In this paper we generalise the inequalities of Schlather and Tawn (2002) for the sets of extremal coefficients and construct bounds that higher order exponent measures need to satisfy to be consistent with lower order exponent measures. Subsequently we construct nonparametric estimators of the exponent measures which impose, through a likelihood-based procedure, the new dependence constraints and provide an improvement on the unconstrained estimators.

math.ST↗

arXiv2011

Extended Generalised Pareto Models for Tail Estimation

The most popular approach in extreme value statistics is the modelling of threshold exceedances using the asymptotically motivated generalised Pareto distribution. This approach involves the selection of a high threshold above which the model fits the data well. Sometimes, few observations of a measurement process might be recorded in applications and so selecting a high quantile of the sample as the threshold leads to almost no exceedances. In this paper we propose extensions of the generalised Pareto distribution that incorporate an additional shape parameter while keeping the tail behaviour unaffected. The inclusion of this parameter offers additional structure for the main body of the distribution, improves the stability of the modified scale, tail index and return level estimates to threshold choice and allows a lower threshold to be selected. We illustrate the benefits of the proposed models with a simulation study and two case studies.

stat.ME↗