The tail empirical and quantile processes

Abstract

In this chapter, we consider a stationary univariate regularly varying times with tail index $\alpha$ and the distribution function of $X_0$ will be denoted by F.

9.7 Bibliographical notes

The estimation of the tail index (or more generally of the extreme value index) is the statistical problem with the longest history in extreme value theory. For an exhaustive list of reference in the case of i.i.d. random variables, see [dHF06].

The fact that consistency of the tail empirical process implies that of order statistics and the Hill estimators was noticed by [RS98]. Consistency of the tail empirical measure for an AR(p) process with heavy-tailed innovation was obtained by [RS97].

Central and functional central limit theorems for the tail and quantile empirical process of i.i.d. data were obtained, among other, by [CCHM86], [Ein92], [Dre98a, Dre98b]. These references use strong approximation techniques to obtain stronger results such as weighted approximations. [Roo95, Roo09] obtained the functional central limit theorem for the tail empirical process under $\alpha$ and $\beta$-mixing. [Dre00, Dre02] strengthened these results by strong approximation techniques for $\eta$-mixing sequences.

The Hill estimator was introduced in [Hil75]. For i.i.d. data, asymptotic normality of the Hill estimator was proved by [Hal82], [DR84]. [HT85], [CM85] among others. [HW84] proved that the Hill estimator is optimal within certain classes of distributions.

For stationary sequences with a regularly varying marginal distribution, the central limit for the Hill estimator was proved under a $\beta$-mixing condition (which applies to finite order moving averages) by [Hsi91b].

In [Dre98a, Dre98b], the Hill and other estimators of the tail index are expressed as functionals of the tail quantile process and the limiting theory for the latter induces the limiting theory for these estimators.

The approach to the Hill estimator presented here differs mainly from the literature in the use of the tail process and condition $\mathcal {S}(r_n, u_n)$. This makes both the assumptions and the results arguably more transparent. The $\beta$-mixing assumption makes the technical parts of the proofs involved with the temporal dependence relatively easy thanks to the coupling property, but some form of mixing is inevitable. It seems that $\mathcal {S}(r_n, u_n)$ first appeared in [Smi92] in the study of the extremal index of Markov chains. A slightly weaker version, called condition R was used in [O’B74], also in relation to the extremal index. The use of this condition in the context of the tail empirical process is due to [KSW18].

Many other estimators of the tail index have been introduced, some with better practical properties than the Hill estimator, some just for the fun of it. Problem 9.5 extends to time series the estimator introduced for i.i.d. random variables by [DEdH89]. Still for i.i.d. data, [CDM85] consider a kernel estimator of the tail index. In [DM98a] its asymptotic normality was proved for infinite order moving averages.

An extremely important part of the literature is entirely ignored here: that which is concerned with practical methods for choosing the number of order statistics used in the statistic under consideration, particularly the Hill estimator. Better choices and bias reduction techniques allow to obtain better rates of convergence and confidence intervals. These methods usually rely on second order conditions which are seldom checkable. The only reference we know of that try to define and study really data-driven methods are [DK98] and [BT15]. The choice of the number or order statistics is related to the marginal distribution F, not to dependence structure and hence we do not discuss it here.