A Note on Nonparametric Estimation of the CTE

The a-level Conditional Tail Expectation (CTE) of a continuous random variable X is defined as its conditional expectation given the event {X>qa} where qa represents its a-level quantile. It is well known that the empirical CTE (the average of the n(l-a) largest order statistics in a sample of size n) is a negatively biased estimator of the CTE. This bias vanishes as the sample size increases, but in small samples can be significant. In this article it is shown that an unbiased nonparametric estimator of the CTE does not exist. In addition, the asymptotic behavior of the bias of the empirical CTE is studied, and a closed form of expression for its first order term is derived. This expression facilitates the study of the behavior of the empirical CTE with respect to the underlying distribution, and suggests and alternative (to the bootstrap) approach to bias correction. The performance of the resulting estimator is assessed via simulation.

Keywords Risk measures