A popular risk measure, conditional value-at-risk (CVaR), is called expected shortfall (ES) in financial applications … for the analysis is established with the quadrangle theory of risk functions. We derived relationships between elements of … CVaR quadrangle and mixed-quantile quadrangle for discrete distributions with equally probable atoms. The deviation in the …

This paper proposes efficient estimators of risk measures in a semiparametric GARCH model defined through moment … when estimating error quantiles. In order to prevent this efficiency loss in quantile estimation, we propose a quantile … estimator based on inverting an empirical likelihood weighted distribution estimator. It is found that the new quantile …