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 CTEnα, 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(CTEnα). From this result it is possible to identify the specific distributions, within some large classes of distributions, that maximize the bias of CTEnα. 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 CTEnα vanishes at the in-between rate of 1/n when F possesses a positive derivative at the αth quantile.