Estimating the multivariate extremal index function

arXiv:0810.1164v2 [stat.AP] 14 Nov 2008 Bernoulli 14(4), 2008, 1027–1064 DOI: 10.3150/08-BEJ145 Estimating the multivariate extremal index function CHRISTIAN Y. ROBERT ENSAE, Timbre J120, 3 avenue Pierre Larousse, 92245 MalakoffCedex, France. E-mail: chrobert@ensae.fr The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asymptotic normality under long-range dependence and moment conditions. The results are illustrated by means of a simulation study. Keywords: cluster-size distributions; exceedance point processes; extreme value theory; multivariate extremal index function 1. Introduction The motivation for this paper comes from an empirical observation that time series from hydrology, meteorology, environmental sciences, finance, etc. are heavy-tailed and clus- tered when extremal events occur. In particular, it has been recognized in recent decades that the model of independent and identically distributed (i.i.d.) Gaussian random vari- ables is inappropriate for modeling extreme returns of risky assets that are observed during a financial crisis. It is important for risk managers to understand the relative be- havior of the various financial risks to which their institutions are exposed in the event of large losses because they have to anticipate the diversification opportunities so that the risks can be balanced by comovements (between risks) or reversal movements in short time intervals (within risks). Although there are well-developed statistical approaches to characterize the cross- sectional dependence structure of extreme returns of risky assets (see, e.g., [14, 20, 23, 34] and the references therein), problems concerning the estimation of their temporal dependence structure have not received much attention. A notable exception is [46], which proposes a specific class of max-stable processes to model simultaneous dependencies between and within financial time series. However, this ad hoc class of processes is not necessarily suitable for any multivariate time series. The multivariate extremal index This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2008, Vol. 14, No. 4, 1027–1064. This reprint differs from the original in pagination and typographic detail. 1350-7265 c⃝ 2008 ISI/BS 1028 C.Y. Robert function, introduced by Nandagopalan [27, 28], is a quantity which allows one to relate the asymptotic distribution of the vector of pointwise maxima of a stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process since it is equal to the reciprocal of the mean number of clustered extremal events. Therefore, it is a specific measure of the temporal dependence structure of the extreme values of the process. It is the aim of this paper to present a general theory for the inference of this function. We extend the block declustering approach introduced in [37] to the case of multivariate stationary processes: we construct pointwise estimators and study their asymptotic prop- erties. Three assumptions are made: (i) there exist moment restrictions on the amount of clustering of extremes; (ii) the number of two-level exceedances converges weakly – an assumption which will guarantee the existence of the asymptotic variance–covariance matrix of the estimators; (iii) a mixing condition weaker than strong mixing is supposed to hold. Under these assumptions, we prove the asymptotic normality of our estimators. More formally, let (Xl = (Xl,1,...,Xl,d))l≥1 be a strictly stationary sequence with sta- tionary distribution function F(x) = P(Xl,i ≤xi,i = 1,...,d), x = (x1,...,xd) ∈Rd, and univariate marginal distributions Fi(x) = P(Xl,i ≤x), i = 1,...,d. We assume that there exists a family of normalizing sequences in Rd, (un(τ) = (un,1(τ1),...,un,d(τd)))n≥1, τ = (τ1,...,τd) ∈(0,∞)d, such that lim n→∞n(1 −Fi(un,i(τ))) = τ for τ > 0,i = 1,..., d, (1.1) and, for some function ˜H :(0,∞)d 7→[0,1], lim n→∞n(1 −F(un(τ))) = −ln ˜H(τ), for τ ∈(0,∞)d. (1.2) A necessary and sufficient condition for the existence of a sequence (un,i(τ))n≥1 which satisfies (1.1) is that limx→xf,i ¯Fi(x)/ ¯Fi(x−) = 1, where xf,i = sup{u :Fi(u) < 1} and ¯Fi = 1 −Fi (see Theorem 1.7.13 in [22]). A natural choice for un,i(τ) is then given by F ← i (1 −τ/n), τ ∈[0,n), where F ← i is the generalised inverse of Fi, that is, F ← i (τ) = inf{x ∈R:Fi(x) ≥τ}. This assumption is weaker than assuming that Fi is in the domain of attraction of an extreme value distribution since the normalization is linear in this case. However, the function ˜G defined by ˜G(τ) = ˜H(τ −1 1 ,...,τ −1 d ) fo

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