Central limit theorem and almost sure results for the empirical estimator of superquantiles/CVaR in the stationary case

Abstract

In this paper, we show that the difference between the empirical estimator and the Conditional value-at-risk can be written as a simple partial sum + a residual term. Starting from this decomposition, we prove a central limit theorem and some almost sure results for the empirical estimator, for a large class of stationary sequences. We also construct a confidence interval with asymptotic level $1-\alpha$, and we study its coverage level through two different sets of simulation.

Keywords:

• Average value-at-risk
• conditional value-at-risk
• expected shortfall
• superquantiles
• stationary sequences
• mixing sequences

2020 Mathematics Subject Classifications:

• 62M10
• 62G30
• 60F05
• 60F15

Disclosure statement

No potential conflict of interest was reported by the author(s).