Ana C. Cebrián

Generalized Pareto Fit to the Society of Actuaries’ Large Claims Database

Abstract This paper discusses a statistical modeling strategy based on extreme value theory to describe the behavior of an insurance portfolio, with particular emphasis on large claims. The strategy is illustrated using the 1991–92 group medical claims database maintained by the Society of Actuaries. Using extreme value theory, the modeling strategy focuses on the “excesses over threshold” approach to fit generalized Pareto distributions. The proposed strategy is compared to standard parametric modeling based on gamma, lognormal, and log-gamma distributions. Extreme value theory outperforms classical parametric fits and allows the actuary to easily estimate high quantiles and the probable maximum loss from the data.

Testing for Concordance Ordering

We propose inference tools to analyse the ordering of concordance of random vectors. The analysis in the bivariate case relies on tests for upper and lower quadrant dominance of the true distribution by a parametric or semiparametric model, i.e. for a parametric or semiparametric model to give a probability that two variables are simultaneously small or large at least as great as it would be were they left unspecified. Tests for its generalisation in higher dimensions, namely analysed. The parametric and semiparametric setting are based on the copula representation for multivariate distribution, which allows for disentangling behaviour of margins and dependence structure. We propose two types of testing procedures for each setting. The first procedure is based on a formulation of the dominance concepts in terms of values taken by random variables, while the second procedure is based on a formulation in terms of probability levels. For each formulation a distance test and an intersection-union test for inequality constraints are developed depending on the definition of null and alternative hypotheses. An empirical illustration is given for US insurance claim data.

Drought Analysis Based on a Marked Cluster Poisson Model

Abstract This paper presents an operational definition of drought events based on an “excess over threshold” approach applied on rainfall series and develops a stochastic model for describing droughts defined in that way. The model consists of a Poisson cluster process to represent drought occurrence and a marked process composed of three series of random variables (duration, deficit, and maximum intensity) to describe drought severity; it is theoretically justified, and adequate procedures to check its validity are suggested and applied on five Spanish rainfall series. Useful parameters for the design and planning of water resource systems, together with their confidence intervals, are estimated from the model.

Risk measures for events with a stochastic duration: an application to drought analysis

Droughts, as many climatic and environmental phenomena, are events with a random duration. In the monitoring and risk management of this type of phenomena, it is important the development of measures of the risk that an ongoing event ends. This work develops a risk measure conditional on the current state of the event, that can be easily updated in real time. The measure is based on the hazard function of the duration of an event, that is modeled as a parametric function of covariates describing the current state of the process. The use of (time-dependent) internal covariates is often required to describe that state, and maximum likelihood methods cannot be used to estimate the model. Therefore, an approach based on partial likelihood functions that permit the inclusion of both external and internal covariates is suggested. This approach is very general but it has the drawback of requiring some programming to be implemented. However, it is proved that for durations with a geometric distribution, an equivalent and easily implemented approach based on generalized linear models can be used to estimate the hazard function. This methodology is applied to develop a risk measure in drought analysis. The approach is exemplified using the drought series from a Spanish location (Huesca) and internal covariates derived from the rainfall series. The whole modeling process is thoroughly described, including the covariate selection procedure and some new validation tools.

Modeling and projecting the occurrence of bivariate extreme heat events using a non-homogeneous common Poisson shock process

A joint model is proposed for analyzing and predicting the occurrence of extreme heat events in two temperature series, these being daily maximum and minimum temperatures. Extreme heat events are defined using a threshold approach and the suggested model, a non-homogeneous common Poisson shock process, accounts for the mutual dependence between the extreme events in the two series. This model is used to study the time evolution of the occurrence of extreme events and its relationship with temperature predictors. A wide range of tools for validating the model is provided, including influence analysis. The main application of this model is to obtain medium-term local projections of the occurrence of extreme heat events in a climate change scenario. Future temperature trajectories from general circulation models, conveniently downscaled, are used as predictors of the model. These trajectories show a generalized increase in temperatures, which may lead to extrapolation errors when the model is used to obtain projections. Various solutions for dealing with this problem are suggested. The results of the fitted model for the temperature series in Barcelona in 1951–2005 and future projections of extreme heat events for the period 2031–2060 are discussed, using three global circulation model trajectories under the SRES A1B scenario.

A bootstrap test of independence between three temporal nonhomogeneous Poisson processes and its application to heat wave modeling

The main contribution of this work is a bootstrap test to check the independence between temporal nonhomogeneous Poisson processes. The test statistic is based on the close point relation, which adapts the crossed nearest neighbour distance ideas of spatial point processes to the case of nonhomogeneous time point processes. Since it is complicated to obtain the probability distribution of the test statistic under the null hypothesis and the parameters of the processes are usually unknown, the

Dynamic Regression Model for Hourly River Level Forecasting Under Risk Situations: an Application to the Ebro River

p

Modeling and forecasting extreme hot events in the central Ebro valley, a continental-Mediterranean area

p value is obtained using a parametric bootstrap approach. A simulation study shows that the size of the test is close to the nominal one. The power is analyzed considering three approaches for generating dependent nonhomogeneous processes and different levels of dependence, and satisfactory results are obtained in all cases. Although the test was initially intended for Poisson processes, it can be applied to any type of point process in one dimension which can be simulated. This test is a valuable tool in the validation analysis of common Poisson shock models. For the bivariate case, the process can be decomposed into three independent Poisson processes, and the assumption of independence between them has to be checked. As an application, the joint modeling of the occurrence process of extreme heat events in daily maximum and minimum temperatures is described.