A central challenge in risk theory is understanding the behavior of an expected cost associated with \(d \ge 1\) heterogeneous risk factors. In [14], the authors introduced the Covariate-Conditional Tail Expectation ( \(\text {CCTE} \) ), a risk measure based on the cumulative distribution function (c.d.f.) which quantifies losses conditional on a given risk scenario by employing c.d.f level sets as risk regions. They also proposed a consistent estimator of the c.d.f.-based CCTE at fixed risk levels. Furthermore, in [3], the CCTE definition was extended to a depth-based framework, rendering the resulting measure direction-free. This paper investigates the asymptotic behavior of the estimators corresponding to both the c.d.f.-based and depth-based versions of the CCTE as the risk level approaches to zero, focusing on the consistency for extreme risks. Several examples of depth functions are analyzed and their respective limitations are discussed.
