No abstract

The conditional tail expectation CTE is an important actuarial risk measure and a useful tool in financial risk assessment. Under the classical assumption that the second moment of the loss variable is finite, the asymptotic normality of the nonparametric CTE estimator has already been established in the literature. The noted result, however, is not applicable when the loss variable follows any distribution with infinite second moment, which is a frequent situation in practice. With a help of extreme-value methodology, in this paper, we offer a solution to the problem by suggesting a new CTE estimator, which is applicable when losses have finite means but infinite variances.

A weighted Gaussian approximation to tail product-limit process for Pareto-like distributions of randomly right-truncated data is provided and a new consistent and asymptotically normal estimator of the extreme value index is derived. A simulation study is carried out to evaluate the finite sample behavior of the proposed estimator.

We make use of the empirical process theory to approximate the adapted Hill estimator, for censored data, in terms of Gaussian processes. Then, we derive its asymptotic normality, only under the usual second-order condition of regular variation. Our methodology allows to relax the assumptions, made in Einmahl et al.

-functionals summarize numerous statistical parameters and actuarial risk measures. Their sample estimators are linear combinations of order statistics (-statistics). There exists a class of heavy-tailed distributions for which the asymptotic normality of these estimators cannot be obtained by classical results. In this paper we propose, by means of extreme value theory, alternative estimators for -functionals and establish their asymptotic normality. Our results may be applied to estimate the trimmed -moments and financial risk measures for heavy-tailed distributions.