Yuri Goegebeur

We consider the estimation of an extreme conditional quantile for a heavy‐tailed distribution in the case of a strictly stationary time series . Here, denotes the conditional quantile function of given and is small, i.e., smaller than , where is the size of the sample on which the estimation is based. In a first step, we propose a conditional extreme value index estimator in that context of time series and establish its weak convergence using the cluster method. Then, in a second step, by using a Weissman‐type construction, we propose an extreme conditional quantile estimator for which we derive also the weak convergence. The performance of our estimators is illustrated on a simulation study and compared with those of alternative estimators already proposed in the literature. Finally, our methodology is applied on a real dataset of log‐returns of the S&P 500 index and market volatility.

We consider bias-corrected estimation of the extreme value index of a Pareto-type distribution in the censoring framework. The initial estimator is based on a Kaplan-Meier integral from which we remove the bias under a second-order framework. This estimator depends on a suitable external estimation of second-order parameters which is also discussed. The weak convergence of the bias-corrected estimator is established. It has the nice property to have the same asymptotic variance as the initial estimator. This nice feature is illustrated on a simulation study where our estimator is compared to alternatives already introduced in the literature. Finally, our methodology is applied on an insurance dataset.

We consider the estimation of the conditional expectation , provided , at extreme levels, where is a strictly stationary time series, its tail quantile function, is a positive integer and is such that . We use the multivariate regular variation framework and consider initially the case of non‐negative time series. A two‐step method is used to propose an estimator of this risk measure: First, by introducing an estimator in the intermediate case, and then, by extrapolating outside the data by a Weissman‐type construction. Under suitable assumptions, we prove the weak convergence of the estimator of this risk measure. Subsequently, we extend our approach to the case of real‐valued time series by using the decomposition of the original time series into the positive and negative parts, and we prove again the weak convergence of the proposed estimator under additional assumptions. The asymptotic variance of this estimator being difficult to approximate, we show the consistency of the multiplier block bootstrap in our context and use it to construct confidence intervals for . Finally, the finite sample properties of the estimator are evaluated with a simulation study, and the methodology is illustrated on a dataset of daily precipitation measurements.

We consider the estimation of the conditional tail moment at extreme levels for the class of Weibull‐type distributions. A two‐step procedure is introduced where in the first stage one estimates the conditional tail moment at an intermediate level, followed by an extrapolation in the second stage. The asymptotic properties of the estimators introduced in the two stages are derived under suitable assumptions. The finite sample properties of the proposed estimator are examined with a simulation experiment. We conclude with two applications on real life data: wind speed measurements collected at an offshore wind farm and air pollution data.

We consider the estimation of the conditional tail moment at extreme levels for the class of Weibull-type distributions. A two-step procedure is introduced where in the first stage one estimates the conditional tail moment at an intermediate level, followed by an extrapolation in the second stage. The asymptotic properties of the estimators introduced in the two stages are derived under suitable assumptions. The finite sample properties of the proposed estimator are examined with a simulation experiment. We conclude with two applications on real life data: wind speed measurements collected at an offshore wind farm and PM_{2.5} air pollution data.

We propose an estimator of the conditional tail moment (CTM) when the data are subject to random censorship. The variable of main interest and the censoring variable both follow a Pareto-type distribution. We establish the asymptotic properties of our estimator and discuss bias-reduction. Then, the CTM is used to estimate, in case of censorship, the premium principle for excess-of-loss reinsurance. The finite sample properties of the proposed estimators are investigated with a simulation study and we illustrate their practical applicability on a dataset of motor third party liability insurance.