Regional extreme value index estimation and a test of tail homogeneity

Abstract

This paper deals with inference on extremes of heavy‐tailed distributions. We assume distribution functions F of Pareto‐type, where the right‐tail behavior of F is characterized by a strictly positive parameter γ, the so‐called extreme value index (EVI). In some applications, observations from closely related variables are available, with possibly identical EVI γ. If these variables are observed for the same time period, a procedure called BEAR estimator has recently been proposed. We modify this approach allowing for different observation periods and pairwise extreme value dependence of the variables. In addition, we present a new test for equality of the EVI. As an application, we discuss regional flood frequency analysis, where we want to combine rather short sequences of univariate observations with very different lengths measured at many gauges for joint inference. We illustrate our findings on peak discharges from 18 river gauges located at the Mulde basin in Germany, which is known for its severe summer floods, and identify relevant heterogeneous tail behavior, which is not detected by other popular methods. Copyright © 2015 John Wiley & Sons, Ltd.

Conditional heavy-tail behavior with applications to precipitation and river flow extremes

“…For instance, we believe that estimation of annual maximal flow distributions based on the block maxima method should be carried out under the assumption of stationary tail behavior. Our work might be extended to regional estimation under the assumption of regional heavytail homogeneity [Kinsvater et al, 2016]. For statistical inference in such a regional setting it is, in contrast to pure local estimation studied here, of practical importance to derive theory under semi-parametric assumptions in order to be able to estimate the dependence between local estimates.…”

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

“…Our Theorem 4 contains Theorem 3.3 and Corollary 3.4 in Dematteo and Clémençon (2016), which are stated under more restrictive conditions, including equality of all tail indices γ j , a multivariate regular variation assumption on X, von Mises conditions on each marginal distribution of X, and a uniform analogue of condition J (R) [note also that Corollary 3.4 in Dematteo and Clémençon (2016) should, with their notation, read i,j = c i c j ν i,j c 1/α i , c 1/α j in the case i < j; as stated, their covariance matrix may fail to be positive semi-definite, for instance for d = 2 and small c 2 . Compare with Proposition 1 in Kinsvater et al (2016)]. Theorem 4 also contains Proposition 1 in Jiang et al (2017), which is limited to the case d = 2 and X (2) = −X (1) .…”

“…where k j = k j (n) → ∞, with k j /n → 0. This theoretical question is addressed in Dematteo and Clémençon (2016) and further discussed in Kinsvater et al (2016) under the assumption γ j = γ for all j ∈ {1, . .…”

“…Such joint convergence results have only been proven very recently by Dematteo and Clémençon (2016), Kinsvater et al (2016) and Hoga (2018). The methods of proof therein rely on advanced theoretical methodologies, namely multivariate vague convergence in Dematteo and Clémençon (2016) and Kinsvater et al (2016) and multivariate empirical process theory in Hoga (2018), as well as on various ad hoc technical conditions.…”

“…Such joint convergence results have only been proven very recently by Dematteo and Clémençon (2016), Kinsvater et al (2016) and Hoga (2018). The methods of proof therein rely on advanced theoretical methodologies, namely multivariate vague convergence in Dematteo and Clémençon (2016) and Kinsvater et al (2016) and multivariate empirical process theory in Hoga (2018), as well as on various ad hoc technical conditions. These joint asymptotic results have been found useful for testing tail homogeneity across marginals of a multivariate heavy-tailed distribution in environmental or financial contexts (see Kinsvater et al (2016) and Hoga (2018)) and constructing improved tail index estimators by pooling (see Section 3.2 in Dematteo and Clémençon (2016)).…”