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 CTE n α , the empirical α -level CTE (the average of the n ( 1 − α ) 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 E ( CTE n α ) . From this result it is possible to identify the specific distributions, within some large classes of distributions, that maximize the bias of CTE n α . 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 α -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 1 / n or vanishes exponentially fast. This is intriguing as the bias of CTE n α vanishes at the in-between rate of 1 / n when F possesses a positive derivative at the α th quantile.