Nonparametric Estimation of the Family of Risk Measures Based on Progressive Type II Censored Data

Tail risk analysis plays a central strategic role in risk management and focuses on the problem of risk measurement in the tail regions of extreme risks. As one crucial task in tail risk analysis for risk management, the measurement of tail risk variability is less addressed in the literature. Neither the theoretical results nor inference methods are fully developed, which results in the difficulty of modeling implementation. Practitioners are then short of measurement methods to understand and evaluate tail risks, even when they have large amounts of valuable data in hand. In this paper, some nonparametric methods of estimation for the class of variability measures among proportional hazards models based on progressively Type-II censored data are derived. We showed some properties of these estimators. Simulation studies have been performed to see the effectiveness of the proposed methods, and a real data set has been analyzed for illustrative purposes. Some well-known variability measures, such as the Gini mean difference, the Wang right tail deviation and the cumulative residual entropy, are, up to a scale factor, in this class.