We incorporate the conditional value-at-risk (CVaR) quantity into a generalized class of Pickands estimators. By introducing CVaR, the newly developed estimators not only retain the desirable properties of consistency, location, and scale invariance inherent to Pickands estimators, but also achieve a reduction in mean squared error (MSE). To address the issue of sensitivity to the choice of the number of top order statistics used for the estimation, and ensure robust estimation, which are crucial in practice, we first propose a beta measure, which is a modified beta density function, to smooth the estimator. Then, we develop an algorithm to approximate the asymptotic mean squared error (AMSE) and determine the optimal beta measure that minimizes AMSE. A simulation study involving a wide range of distributions shows that our estimators have good and highly stable finite-sample performance and compare favorably with the other estimators.

翻译：本文将条件风险价值（CVaR）量纳入广义Pickands估计量类中。通过引入CVaR，新开发的估计量不仅保留了Pickands估计量固有的相合性、位置与尺度不变性等优良性质，还实现了均方误差（MSE）的降低。为解决估计过程对所用顶端次序统计量个数选择的敏感性问题，并确保实际应用中至关重要的稳健估计，我们首先提出一种贝塔测度——即修正的贝塔密度函数——以平滑估计量。随后，我们开发了一种算法来近似渐近均方误差（AMSE），并确定使AMSE最小化的最优贝塔测度。涵盖广泛分布的模拟研究表明，我们的估计量具有良好的、高度稳定的有限样本性能，并优于其他估计量。