Tail Mean Estimation is More Efficient than Tail Median: Evidence From the Exponential Power Distribution

We investigate the relative efficiency of the empirical “tail median” vs. ”tail mean” as estimators of location under random sampling from the exponential power distribution (EPD). By considering appropriate probabilities so that the quantile of the untruncated EPD (population tail median) and mean of the left-truncated EPD (population tail mean) coincide, limiting results are established concerning the ratio of asymptotic variances of the corresponding estimators. The most remarkable finding is that in the limit of the right tail, the asymptotic variance of the tail median is approximately 36% larger than that of the tail mean, irrespective of the EPD shape parameter. This discovery has important repercussions for quantitative risk management practitioners, where the tail median and tail mean correspond to value-at-risk and expected shortfall, respectively. From a purely academic standpoint, the findings also offer a generalized solution to the age-old statistical quandary: which of the sample median vs. sample mean is the most efficient estimator of centrality?