On the study of extremes with dependent random right-censoring

The study of extremes in missing data frameworks is a recent developing field. In particular, the randomly right-censored case has been receiving a fair amount of attention in the last decade. All studies on this topic, however, essentially work under the usual assumption that the variable of interest and the censoring variable are independent. Furthermore, a frequent characteristic of estimation procedures developed so far is their crucial reliance on particular properties of the asymptotic behaviour of the response variable Z (that is, the minimum between time-to-event and time-to-censoring) and of the probability of censoring in the right tail of Z. In this paper, we focus instead on elucidating this asymptotic behaviour in the dependent censoring case, and, more precisely, when the structure of the dependent censoring mechanism is given by an extreme value copula. We then draw a number of consequences of our results, related to the asymptotic behaviour, in this dependent context, of a number of estimators of the extreme value index of the random variable of interest that were introduced in the literature under the assumption of independent censoring, and we discuss more generally the implications of our results on the inference of the extremes of this variable

Effects of extreme surges.

Extreme value analysis of sea levels is an essential component of risk analysis and protection strategy for many coastal regions. Since the tidal component of the sea level is deterministic, it is the stochastic variation in extreme surges that is the most important to model. Historically, this modelling has been accomplished by fitting classical extreme value models to series of annual maxima data. Recent developments in extreme value modelling have led to alternative procedures that make better use of available data, and this has led to much refined estimates of extreme surge levels. However, one aspect that has been routinely ignored is seasonality. In an earlier study we identified strong seasonal effects at one of the number of locations along the eastern coastline of the United Kingdom. In this article, we discuss the construction and inference of extreme value models for processes that include components of seasonality in greater detail. We use a point process representation of extreme value behaviour, and set our inference in a Bayesian framework, using simulation-based techniques to resolve the computational issues. Though contemporary, these techniques are now widely used for extreme value modelling. However, the issue of seasonality requires delicate consideration of model specification and parameterization, especially for efficient implementation via Markov chain Monte Carlo algorithms, and this issue seems not to have been much discussed in the literature. In the present paper we make some suggestions for model construction and apply the resultant model to study the characteristics of the surge process, especially in terms of its seasonal variation, on the eastern UK coastline. Furthermore, we illustrate how an estimated model for seasonal surge can be combined with tide records to produce return level estimates for extreme sea levels that accounts for seasonal variation in both the surge and tidal processes

Estimation of return periods for extreme sea levels: a simplified empirical correction of the joint probabilities method with examples from the French Atlantic coast and three ports in the southwest of the UK

Ocean Dynamics DOI 10.1007/s10236-006-0096-8 Paolo Antonio Pirazzoli . Alberto Tomasin Estimation of return periods for extreme sea levels: a simplified empirical correction of the joint probabilities method with examples from the French Atlantic coast and three ports in the southwest of the UK Accepted: 8 November 2006 # Springer-Verlag 2007 Abstract The joint probability method (JPM) to estimate the probability of extreme sea levels (Pugh and Vassie, Extreme sea-levels from tide and surge probability. Proc. 16th Coastal Engineering Conference, 1978, Hamburg, American Society of Civil Engineers, New York, pp 911– 930, 1979) has been applied to the hourly records of 13 tide-gauge stations of the tidally dominated Atlantic coast of France (including Brest, since 1860) and to three stations in the southwest of the UK (including Newlyn, since 1916). The cumulative total length of the available records (more than 426 years) is variable from 1 to 130 years when individual stations are considered. It appears that heights estimated with the JPM are almost systematically greater than the extreme heights recorded. Statistical analysis shows that this could be due: (1) to surge–tide interaction (that may tend to damp surge values that occur at the time of the highest tide levels), and (2) to the fact that major surges often occur in seasonal periods that may not correspond to those of extreme astronomical tides.We have determined at each station empirical ad hoc correction coefficients that take into account the above two factors separately, or together, and estimated return periods for extreme water levels also at stations where only short records are available. For seven long records, for which estimations with other computing methods (e.g. generalized extreme value [GEV] distribution and Gumbel) can be attempted, average estimations of extreme values appear slightly overestimated in relation to the actual maximum records by the uncorrected JPM (+16.7±7.2 cm), and by the Gumbel method alone (+10.3±6.3 cm), but appear closer to the reality with the GEV distribution (−2.0± 5.3 cm) and with the best-fitting correction to the JPM (+2.9±4.4 cm). Because the GEV analysis can hardly be extended to short records, it is proposed to apply at each station, especially for short records, the JPM and the sitedependent ad hoc technique of correction that is able to give the closest estimation to the maximum height actually recorded. Extreme levels with estimated return times of 10, 50 and 100 years, respectively, are finally proposed at all stations. Because astronomical tide and surges have been computed (or corrected) in relation to the yearly mean sea level, possible changes in the relative sea level of the past, or foreseeable in the future, can be considered separately and easily added to (or deduced from) the extremes obtained. Changes in climate, on the other hand, may modify surge and tide distribution and hence return times of extreme sea levels, and should be considered separately. Keywords Tide gauge . Sea level . Extreme values . Return period . Atlantic coast . France . UK 1 Introduction Most methods usually employed to estimate return periods of extreme values for hydrological or meteorological datasets (extremes per block, threshold method, annual maxima [Gumbel] method) are based on a number of assumptions: (1) that we deal with statistical variates; (2) that the initial distribution from which the extremes have been drawn, and its parameters, remains constant from one Responsible editor: Roger Proctor Parts of this paper have been presented orally at the session “Geophysical extremes: scaling aspects and modern statistical approaches” of the EGU General Assembly, Vienna, 2–6 April 2006. P. A. Pirazzoli (*) Laboratoire de Géographie Physique, Centre National de la Recherche Scientifique (CNRS), 1 Place Aristide Briand, 92195 Meudon Cedex, France e-mail: [email protected] A. Tomasin Università di Venezia