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We propose a nonparametric robust estimator for the tail index of a conditional Pareto-type distribution in the presence of censoring and random covariates. The censored distribution is also of Pareto-type and the index is estimated locally within a narrow neighbourhood of the point of interest in the covariate space using the minimum density power divergence method. The main asymptotic properties of our robust estimator are derived under mild regularity conditions and its finite sample performance is illustrated on a small simulation study. A real data example is included to illustrate the practical applicability of the estimator.

We introduce a flexible class of kernel type estimators of a second order parameter appearing in the multivariate extreme value framework. Such an estimator is crucial in order to construct asymptotically unbiased estimators of dependence measures, as e.g. the stable tail dependence function. We establish the asymptotic properties of this class of estimators under suitable assumptions. The behaviour of some examples of kernel estimators is illustrated by a simulation study in which they are also compared with a benchmark estimator of a second order parameter recently introduced in the literature. [ABSTRACT FROM AUTHOR]

In multivariate extreme value statistics, the estimation of probabilities of extreme failure sets is an important problem, with practical relevance for applications in several scientific disciplines. Some estimators have been introduced in the literature, though so far the typical bias issues that arise in application of extreme value methods and the non-robustness of such methods with respect to outliers were not addressed. We introduce a bias-corrected and robust estimator for small tail probabilities. The estimator is obtained from a second order model that is fitted to properly transformed bivariate observations by means of the minimum density power divergence technique. The asymptotic properties are derived under some mild regularity conditions and the finite sample performance is evaluated through an extensive simulation study. We illustrate the practical applicability of the method on a dataset from the actuarial context.

We introduce a nonparametric regression estimator for a tail heaviness parameter in an integrated conditional Pareto–Weibull-type model. The estimator is based on local log excesses over a high random threshold. Asymptotic properties are derived under proper regularity conditions. [ABSTRACT FROM AUTHOR]

Abstract: A robust and asymptotically unbiased extreme quantile estimator is derived from a second order Pareto-type model and its asymptotic properties are studied under suitable regularity conditions. The finite sample properties of the proposed estimator are investigated with a small simulation experiment. [Copyright &y& Elsevier]

Abstract: We present a method for estimating the endpoint of a unidimensional sample when the distribution function belongs to the Weibull-max domain of attraction. The approach relies on transforming the variable of interest and then using high order moments of the positive variable obtained this way. It is assumed that the order of the moments goes to infinity. We give conditions on the rate of divergence to get the weak and strong consistency as well as the asymptotic normality of the estimator. The good performance of the estimator is illustrated on some finite sample situations. [Copyright &y& Elsevier]

The P.O.T. (Peaks-Over-Threshold) approach consists of using the Generalized Pareto Distribution (GPD) to approximate the distribution of excesses over a threshold. We use the probability-weighted moments to estimate the parameters of the approximating distribution. We study the asymptotic behaviour of these estimators (in particular their asymptotic bias) and also the functional bias of the GPD as an estimate of the distribution function of the excesses. We adapt penultimate approximation results to the case where parameters are estimated.

In order to construct confidence sets for a marginal density fof a strictly stationary continuous time process observed over the time interval [0, T], it is necessary to have at one's disposal a Central Limit Theorem for the kernel density estimator fT. In this paper we address the question of nonparametric estimation of the asymptotic variance of TfT, an unknown quantity dependent on f. We construct two estimators and study their asymptotic properties.

In the usual right-censored data situation, let f n, n∈ N, denote the convolution of the Kaplan–Meier product limit estimator with the kernels an−1K(·/ an), where Kis a smooth probability density with bounded support and an→0. That is, fnis the usual kernel density estimator based on Kaplan–Meier. Let f̄ndenote the convolution of the distribution of the uncensored data, which is assumed to have a bounded density, with the same kernels. For each  n, let Jndenote the half line with right end point Zn(1− εn), n− an, where εn→0 and, for each  m, Zm, nis the mth order statistic of the censored data. It is shown that, under some mild conditions on anand  εn, supJ n|f n(t)− f̄n(t)| converges a.s. to zero as n→∞ at least as fast as | log(a n∧ε n)|/(na nε n) . For εn=constant, this rate compares, up to constants, with the exact rate for fixed intervals.

In the present paper we develop second-order theory using the subsample bootstrap in the context of Pareto index estimation. We show that the bootstrap is not second-order accurate, in the sense that it fails to correct the first term describing departure from the limit distribution. Worse than this, even when the subsample size is chosen optimally, the error between the subsample bootstrap approximation and the true distribution is often an order of magnitude larger than that oi tue asymptotic approximation. To overcome this deficiency, we show that an extrapolation method, based quite literally on a mixture of asymptotic and subsample bootstrap methods, can lead to second-order correct confidence intervals for the Pareto index.

Let X1, ..., Xnbe an i.i.d. sequence of random variables, from an unknown distribution F, and X1W, ... XnWbe a sample from $$F_{W,n} $$ , the weighted empirical distribution function of X1, ..., Xn. We define the order statistics X1,nW≤ ... ≤ Xn,nWof X1W, ..., XnW. Under suitable assumptions on weights, we study the influence of the maxima in the construction of limit theorems. We choose a resample size m(n)and we derive conditions on m(n)for the “in probability” and “with probability 1” consistency of Xm(n),m(n)W. The presence of weights has an influence on the resample size and requires the use of new tools. When Xn,nis in the domain of attraction of an extreme value distribution, m(n) → ∞, and $$\frac{{m(n)}}{{\sqrt n }} \to 0$$ , as n → ∞, all our results hold.

The marginal expected shortfall is an important risk measure in finance and actuarial science, which has been extended recently to the case where the random variables of main interest are observed together with a covariate. This leads to the concept of conditional marginal expected shortfall for which an estimator is proposed allowing extrapolation outside the data range. The main asymptotic properties of this estimator have been established, using empirical processes arguments combined with the multivariate extreme value theory. The finite sample behavior of the proposed estimator is evaluated with a simulation experiment, and the practical applicability is illustrated on vehicle insurance customer data.