Estimation of the marginal expected shortfall under asymptotic independence

Abstract

We study the asymptotic behavior of the marginal expected shortfall when the two random variables are asymptotic independent but positively associated, which is modeled by the so‐called tail dependent coefficient. We construct an estimator of the marginal expected shortfall, which is shown to be asymptotically normal. The finite sample performance of the estimator is investigated in a small simulation study. The method is also applied to estimate the expected amount of rainfall at a weather station given that there is a once every 100 years rainfall at another weather station nearby.

Hidden regular variation, copula models, and the limit behavior of conditional excess risk measures

“…The asymptotic behavior of the MME and the MES under hidden regular variation were obtained in Das and Fasen-Hartmann [15]. Under these constraints consistent estimators for MME and MES were derived; moreover Cai and Musta [8] have also shown asymptotic normality for MES. In this section we present a variety of examples of model classes satisfying the assumptions of Das and Fasen-Hartmann [15] and relate these assumptions in particular, to copula models.…”

“…In particular, we show that the asymptotic behavior of MES(p) depends only on the tail of Z 1 and the tail copula; neither the presence of hidden regular variation nor the tail of Z 2 have any influence on the limit. We compare the conclusions for MES(p) with those of Cai and Musta [8]. Finally, we provide several examples for models satisfying our assumptions.…”

“…for some α > 1 and constant K 1 ∈ (0, ∞). Suppose C has upper tail order pair (κ, τ) with κ ≥ τ ≥ 1 and some slowly varying function ℓ at 0 with lim s↓0 s κ−1 ℓ(s) = 0 satisfying (7) and (8) Proof. Similar to the proof of Theorem 4.8, the proof here follows easily if we show that conditions (E) and (F) are equivalent.…”

“…The asymptotic tail behavior of MES was discussed under the assumption of regular variation and asymptotic tail dependence in Cai et al [9], Hua and Joe [22], and under asymptotic tail independence in Cai and Musta [8], Das and Fasen-Hartmann [15]. Such risk metrics have also been studied in a time-series context in Kulik and Soulier [32], and under a copula-framework in Hua and Joe [24].…”

“…Interestingly, in the asymptotically tail-independent case these risk measures may have different rates of convergence or even converge to a constant, e.g., for independent random variables Z 1 , Z 2 we have MES(p) = E(Z 1 ). The asymptotic behavior of MME and MES for a hidden regularly varying random vector Z = (Z 1 , Z 2 ) has been investigated in Das and Fasen-Hartmann [15], furthermore, consistent estimators for these risk measures based on methods from extreme value theory have been proposed; for asymptotic normality of MES see Cai and Musta [8]. On the other hand, the asymptotic behavior of the risk measures as defined in (4) are not particularly well-studied, especially in the context of heavy-tailed asymptotically tail independent risks.…”