Tail expectile process and risk assessment

Abstract

Expectiles define a least squares analogue of quantiles. They are determined by tail expectations rather than tail probabilities. For this reason and many other theoretical and practical merits, expectiles have recently received a lot of attention, especially in actuarial and financial risk management. Their estimation, however, typically requires to consider non-explicit asymmetric least squares estimates rather than the traditional order statistics used for quantile estimation. This makes the study of the tail expectile process a lot harder than that of the standard tail quantile process. Under the challenging model of heavy-tailed distributions, we derive joint weighted Gaussian approximations of the tail empirical expectile and quantile processes. We then use this powerful result to introduce and study new estimators of extreme expectiles and the standard quantile-based expected shortfall, as well as a novel expectile-based form of expected shortfall. Our estimators are built on general weighted combinations of both top order statistics and asymmetric least squares estimates. Some numerical simulations and applications to actuarial and financial data are provided.

ExpectHill estimation, extreme risk and heavy tails

“…Second, Theorem 2 in Daouia et al [15] unravels the discrepancy between the tail empirical expectile process and its population counterpart. As these two theorems constitute the basic theoretical tools for our asymptotic analysis in the present paper, they are briefly described below in Proposition 1 along with the statistical model in Section 2.…”

“…Similarly to this intuitive tail conditional expectation, Taylor [45] has introduced and used the expectile-based form ErY |Y ą ξ τ s as the basis for estimating the standard quantilebased measure ErY |Y ą q τ s. Given that both conditional expectations ErY |Y ą q τ s and ErY |Y ą ξ τ s are not coherent risk measures in general, Daouia et al [15] have suggested to estimate the coherent ES form QES τ on the basis of its expectile-based analogue XES τ :" 1 1´τ…”

“…obtained by substituting the expectile ξ t in place of the quantile q t in QES τ . This definition is more convenient than ErY |Y ą ξ τ s as it induces a proper coherent risk measure (see Proposition 2 in Daouia et al [15]).…”

“…Yet, we have initiated a satisfactory solution to this problem in Daouia et al [13] by proposing intermediate and extreme expectile estimators and developing their asymptotic theory. Very recently, we have come up in Daouia et al [15] with powerful approximations of the tail empirical expectile process. First, Theorem 1 in Daouia et al [15] derives an explicit joint asymptotic Gaussian representation of the tail expectile and quantile processes.…”

“…Very recently, we have come up in Daouia et al [15] with powerful approximations of the tail empirical expectile process. First, Theorem 1 in Daouia et al [15] derives an explicit joint asymptotic Gaussian representation of the tail expectile and quantile processes.…”