Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles

Abstract

We investigate the asymptotic behavior of the (relative) extrapolation error associated with some estimators of extreme quantiles based on extreme-value theory. It is shown that the extrapolation error can be interpreted as the remainder of a first order Taylor expansion. Conditions are then provided such that this error tends to zero as the sample size increases. Interestingly, in case of the so-called Exponential Tail estimator, these conditions lead to a subdivision of Gumbel maximum domain of attraction into three subsets. In contrast, the extrapolation error associated with Weissman estimator has a common behavior over the whole Fréchet maximum domain of attraction. First order equivalents of the extrapolation error are then derived showing that Weissman estimator may lead to smaller extrapolation errors than the Exponential Tail estimator on some subsets of Gumbel maximum domain of attraction. The accuracy of the equivalents is illustrated numerically and an application on real data is also provided.

An extreme quantile estimator for the log-generalized Weibull-tail model

“…Let us also note that assuming θ = 0 improves the results only on the strict Weibull distribution, the results ofQ [3] n (β n ) being disappointing for other Weibull tail-distributions such as Gaussian or Gamma. Similarly, assuming θ > 0 improves the results only on the Gamma distribution, the results ofQ [2] n (β n ) are not convincing on other distributions. This phenomenon indicates thatθ n (βn) (solid line), Q [2] n (βn) (dotted line),Q [3] n (βn) (dash-dotted line) andQ n (βn) (solid line), Q [2] n (βn) (dotted line),Q [3] n (βn) (dash-dotted line) andQ [4] n (βn) (dashed line) as functions of kn for βn = n −2 and n = 5000. n (βn) (solid line), Q [2] n (βn) (dotted line),Q [3] n (βn) (dash-dotted line) andQ [4] n (βn) (dashed line) as functions of kn for βn = n −2 and n = 5000. n (βn) (solid line), Q [2] n (βn) (dotted line),Q [3] n (βn) (dash-dotted line) andQ [4] n (βn) (dashed line) as functions of kn for βn = n −2 and n = 5000. n (βn) (solid line), Q [2] n (βn) (dotted line),Q [3] n (βn) (dash-dotted line) andQ n (βn) (solid line), Q [2] n (βn) (dotted line),Q [3] n (βn) (dash-dotted line) andQ [4] n (βn) (dashed line) as functions of kn for βn = n −2 and n = 5000. n (βn) (solid line),Q [2] n (βn) (dotted line),Q [3] n (βn) (dash-dotted line) andQ [4] n (βn) (dashed line) as functions of kn for βn = n −2 and n = 5000. n (βn) (solid line),Q [2] n (βn) (dotted line),Q [3] n (βn) (dash-dotted line) andQ [4] n (βn) (dashed line) as functions of kn for βn = n −2 and n = 5000.…”

“…One can then easily show that such a tail parameter τ coincides with the index θ of extended regular variation in the situation where θ ∈ [0, 1]. In terms of Maximum Domain of Attraction (MDA), the following result has been established in [2,Proposition 4]: Lemma 1. Assume F is twice differentiable.…”

“…The situation θ > 1 which does not correspond to any domain of attraction is sometimes referred to as super-heavy tails, see for instance [3]. The following examples are taken from [2,Proposition 3]: Example 1. Let x * := sup{x ≥ 1, F (x) < 1} be the endpoint of F .…”

“…n , see (8) and (7). The resulting estimatorQ [2] n (β n ) should perform well for estimating extreme quantiles from distributions with associated θ ≥ 0.…”

“…Let (k n ) be an intermediate sequence such that k n → ∞ and ln(k n )/ ln(n) → 0 as n → ∞. For J 1 , J 2 ∈ N \ {0} and for all ζ (1) ∈ (−∞, 1) J1 , ζ (2) ∈ (−∞, 1) J2 , the random vector…”