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 … estimator is uniformly more efficient than the simple empirical quantile and a quantile estimator based on normalized residuals …

CVaR quadrangle and mixed-quantile quadrangle for discrete distributions with equally probable atoms. The deviation in the … two sets of parameters for the mixed-quantile quadrangle. For the first set of parameters, the minimization of error from … the CVaR quadrangle is equivalent to the minimization of the Rockafellar error from the mixed-quantile quadrangle …