On tail index estimation based on multivariate data

Abstract

This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, i.e. of which Pareto-like marginals share the same tail index. A multivariate Central Limit Theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common tail index α, is established under mild conditions. Motivated by the statistical analysis of extremal spatial data in particular, we introduce the concept of (standard) heavy-tailed random field of tail index α and show how this limit result can be used in order to build an estimator of α with small asymptotic mean squared error, through a proper convex linear combination of the coordinates. Beyond asymptotic results, simulation experiments illustrating the relevance of the approach promoted are also presented.

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

“…The benefit of writing this is that while showing directly the joint convergence of the random pair on the left-hand side is difficult and appears to require advanced theoretical arguments (see Dematteo and Clémençon (2016) and Hoga (2018)), the convergence of the right-hand side is much easier to obtain since it is nothing but a pair of (ratios of) sums of independent and identically distributed random variables. We will return to this in Section 3 to show how this observation leads to conceptually simple proofs of the joint asymptotic normality of several Hill estimators.…”

“…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)).…”

“…The class of average excesses we consider includes the Hill estimator, and can be modified in a very simple way to encompass Conditional Tail Moments (CTMs). We show in particular how the results of Dematteo and Clémençon (2016), on the joint asymptotic normality of marginal Hill estimators for a random vector with heavy-tailed marginal distributions, can be recovered and generalised under weaker assumptions and by elementary techniques. We shall then highlight a couple of applications of this joint asymptotic normality result, including to the obtention of the asymptotic properties of the tail index estimator introduced by Gardes and Stupfler (2015) when the variable of interest is randomly right-truncated.…”