Bounds for the bias of the empirical CTE

The Conditional Tail Expectation (CTE) is gaining an increasing level of attention as a measure of risk. It is known that nonparametric unbiased estimators of the CTE do not exist, and that , the empirical [alpha]-level CTE (the average of the n(1-[alpha]) largest order statistics in a random sample of size n), is negatively biased. In this article, we show that increasing convex order among distributions is preserved by . From this result it is possible to identify the specific distributions, within some large classes of distributions, that maximize the bias of . This in turn leads to best possible bounds on the bias under various sets of conditions on the sampling distribution F. In particular, we show that when the [alpha]-level quantile is an isolated point in the support of a non-degenerate distribution (for example, a lattice distribution) then the bias is either of the order or vanishes exponentially fast. This is intriguing as the bias of vanishes at the in-between rate of 1/n when F possesses a positive derivative at the [alpha]th quantile.