Identifying groups of variables with the potential of being large simultaneously

Abstract

Identifying groups of variables that may be large simultaneously amounts to finding out which joint tail dependence coefficients of a multivariate distribution are positive. The asymptotic distribution of a vector of nonparametric, rank-based estimators of these coefficients justifies a stopping criterion in an algorithm that searches the collection of all possible groups of variables in a systematic way, from smaller groups to larger ones. The issue that the tolerance level in the stopping criterion should depend on the size of the groups is circumvented by the use of a conditional tail dependence coefficient. Alternatively, such stopping criteria can be based on limit distributions of rank-based estimators of the coefficient of tail dependence, quantifying the speed of decay of joint survival functions. A performance score calculated by ten-fold cross-validation allows the user to select one among the various algorithms and set its tuning parameters in a data-driven way.

Appendices

Appendix A: Proofs

Proof of Proposition 1

For ∅ ≠ a ⊂{1,…, d} and x ∈ [0, ∞)^a, put

$$\begin{array}{@{}rcl@{}} L_{{a}}(\boldsymbol{x}) &=& \{ \boldsymbol{y} \in [0, \infty]^{d} \mid \exists j \in {a} : y_{j} < x_{j} \}, \\ R_{{a}}(\boldsymbol{x}) &=& \{ \boldsymbol{y} \in [0, \infty]^{d} \mid \forall j \in {a} : y_{j} < x_{j} \}. \end{array} $$

If a = {1,…, d}, then just write L rather than L_{{1,…, d}}. Note that L_a(x) = L(xe_a) with and that L_{j}(x_j) = R_{j}(x_j) and thus W(L_{j}(x_j)) = W_{j}(x_j). Einmahl et al. (2012, Theorem 4.6) show that, in the space ℓ^∞([0, T]^d) and under Conditions 1, 2 and 3, we have weak convergence

$$\sqrt{k}\{ \widehat{\ell}(\boldsymbol{x}) - \ell(\boldsymbol{x}) \} \rightsquigarrow W(L(\boldsymbol{x})) - \sum\limits_{j = 1}^{d} \partial \ell_{j}(\boldsymbol{x}) W_{\{j\}}(x_{j}) $$

as n →∞. Here, we have taken a version of the Gaussian process W such that the trajectories x↦W(L(x)) are continuous almost surely.

As in Eq. 7, we have, for ∅≠a ⊂{1,…, d} and x ∈ [0, ∞)^a, the identity

$$\widehat{r}_{{a}}(\boldsymbol{x}) = \sum\limits_{\emptyset \ne {b} \subset {a}} (-1)^{|{b}|+ 1} \widehat\ell(\boldsymbol{x}_{b} \boldsymbol{e}_{b} ) $$

where x_b = (x_j)_j∈b. Hence, we can view the vector $(\sqrt {k}(\widehat {r}_{{a}} - r_{{a}} ))_{\emptyset \ne {a} \subset \{1,\ldots ,d\}}$ as the result of the application to $\sqrt {k}(\widehat {\ell }-\ell )$ of a bounded linear map from ℓ^∞([0, T]^d) to ${\prod }_{\emptyset \ne {a} \in \{1,\ldots ,d\}} \ell ^{\infty }([0, T]^{{a}})$. By the continuous mapping theorem, we obtain, in the latter space, the weak convergence

Here we used ℓ(x_be_b) = ℓ_b(x_b).

The set-indexed process W satisfies the remarkable property that W(A ∪ B) = W(A) + W(B) almost surely whenever A and B are disjoint Borel sets of [0, ∞]^d ∖{∞} that are bounded away from ∞: indeed, Eq. 9 implies ${\mathbb {E}}[\{W(A \cup B) - W(A) - W(B)\}^{2}] = 0$. It follows that the trajectories of W obey the inclusion-exclusion formula, so that, for ∅ ≠ a ⊂{1,…, d} and x ∈ [0, ∞)^a, we have, almost surely,

$$\begin{array}{@{}rcl@{}} \sum\limits_{\emptyset \ne {b} \subset {a}} (-1)^{|{b}|+ 1} W(L_{b}(\boldsymbol{x}_{b})) &=& \sum\limits_{\emptyset \ne {b} \subset {a}} (-1)^{|{b}|+ 1} W\left( {\bigcup}_{j\in{b}} R_{\{j\}}(x_{j}) \right) \\ &=& W\left( {\bigcap}_{j\in{a}} R_{\{j\}}(x_{j}) \right) = W(R_{{a}}(\boldsymbol{x})) = W_{{a}}(\boldsymbol{x}). \end{array} $$

We can make this hold true almost surely jointly for all such a and x: first, consider points x with rational coordinates only and then consider a version of W by extending W_a to points x with general coordinates via continuity. Similarly, since W_{j}(0) = W(∅) = 0 almost surely, we have

We have thus shown weak convergence as stated in Eq. 10. □

Proof of Corollary 1

The weak convergence statement (13) is a special case of Eq. 10: set x = 1_a. The covariance formula (14) follows from the fact that

$$\begin{array}{@{}rcl@{}} {\mathbb{E}}[W_{{a}}(\boldsymbol{1}_{{a}}) W_{{a}^{\prime}}(\boldsymbol{1}_{{a}^{\prime}})] &=& {\Lambda} (\{ \boldsymbol{y} \in [0, \infty]^{d} \mid \forall i \in {a} \cup {a}^{\prime} : y_{i} < 1 \} ) \\ &=& \mu(\{ \boldsymbol{u} \in [0, \infty)^{d} \mid \forall i \in {a} \cup {a}^{\prime} : u_{i} > 1 \} ) = \chi_{{a}\cup{a}^{\prime}}; \end{array} $$

the first equality follows from Eq. 9 and the last one from Eq. 3. We obtain Eq. 14 by expanding G_a = Z_a(1_a) using Eq. 10 and working out ${\mathbb {E}}[ G_{{a}} G_{{a}^{\prime }} ]$ with the above identity. □

Proof of Proposition 2

Let a = {a₁,…, a_S}⊂{1,…, d} with S = |a|≥ 2 and such that μ(Δ_a) > 0. In view of Eq. 12, we have κ_a = g_a(τ_a) and $\widehat {\kappa }_{{a}} = g_{{a}}(\widehat {\tau }_{{a}} )$ where τ_a = (χ_a, χ_a∖a1,…, χ_a∖aS), $\widehat \tau _{{a}} = (\widehat \chi _{{a}}$, $\widehat \chi _{{a}\setminus {a}_{1}}$, …, $\widehat \chi _{{a}\setminus {a}_{S}})$, and

$$ g_{{a}}(x_{0}, x_{1}, \ldots, x_{S}) = \frac{x_{0}}{{\sum}_{j = 1}^{S} x_{j} - (S-1) x_{0}}, \qquad x\in[0,\infty)^{1+S}. $$

(29)

Let ∇g_a(x) denote the gradient vector of g_a evaluated x and let 〈⋅,⋅〉 denote the scalar product in Euclidean space. Proposition 1 combined with the delta method as in van der Vaart (1998, Theorem 3.1) gives, as n →∞,

$$\begin{array}{@{}rcl@{}} \sqrt{k} (\widehat\kappa_{{a}} - \kappa_{{a}} ) = \sqrt{k} \{ g_{{a}}(\widehat{\tau}_{{a}} ) - g_{{a}}(\tau_{{a}} ) \} &=& \left\langle \nabla g_{{a}}(\tau_{{a}}), \sqrt{k} (\widehat{\tau}_{{a}} - \tau_{{a}}) \right\rangle + \mathrm{o}_{{\mathbb{P}}}(1) \\ &\rightsquigarrow& \left\langle \nabla g_{{a}}(\tau_{{a}}), (G_{{a}},G_{{a}\setminus{a}_{1}},\ldots,G_{{a}\setminus{a}_{S}}) \right\rangle, \end{array} $$

the weak convergence holding jointly in a by Slutsky’s lemma and Proposition 1. The partial derivatives of g_a are

$$\begin{array}{@{}rcl@{}} \frac{\partial g}{\partial x_{0}}(x ) &=& \frac{{\sum}_{j = 1}^{S} x_{j} }{\left\{ {\sum}_{j = 1}^{S} x_{j} - (S-1) x_{0} \right\}^{2}}, \\ \frac{\partial g}{\partial x_{j}}(x ) &=& \frac{- x_{0}}{\left\{ {\sum}_{j = 1}^{S} x_{j} - (S-1) x_{0} \right\}^{2}}, \qquad j = 1, \ldots, S. \end{array} $$

Evaluating these at x = τ_a and using ${\sum }_{j \in {a}} \chi _{{a} \setminus j} - (S-1) \chi _{{a}} = \mu ({\Delta }_{{a}})$ as in Eqs. 11 and 12, we find that

$$\left\langle \nabla g_{{a}}(\tau_{{a}}), (G_{{a}},G_{{a}\setminus{a}_{1}},\ldots,G_{{a}\setminus{a}_{S}}) \right\rangle = \mu({\Delta}_{{a}})^{-2} \left\{ \left( {\sum}_{j\in{a}}\chi_{{a}\setminus j} \right) G_{{a}} - \chi_{{a}} {\sum}_{j\in{a}} G_{{a}\setminus j} \right\}, $$

in accordance to the right-hand side in Eq. 16.

To calculate the asymptotic variance $\sigma _{\kappa ,{a}}^{2}$, we introduce a few abbreviations: we write R_b = R_b(1_b) and $W_{b}^{\cap } = W_{b}(\boldsymbol {1}_{b}) = W(R_{b})$ for ∅≠b ∈{1,…, d} and we put W_j = W_{j}(1) for j = 1,…, d, so that $G_{{a}} = W_{{a}}^{\cap } - {\sum }_{j\in {a}}\dot {\chi }_{j,{a}} W_{j}$. We find

$$\begin{array}{@{}rcl@{}} H_{{a}} &=& \left( \sum\limits_{i\in{a}}\chi_{{a}\setminus i} \right) G_{{a}} - \chi_{{a}} \sum\limits_{i\in{a}} G_{{a}\setminus i} \\ &=& \left( \sum\limits_{i\in{a}}\chi_{{a}\setminus i}\right) \left( W^{\cap}_{{a}} - \sum\limits_{j\in{a}}\dot{\chi}_{j,{a}}W_{j}\right) - \chi_{{a}} \sum\limits_{i\in{a}} \left( W^{\cap}_{{a}\setminus i} - \sum\limits_{j\in{a}\setminus i} \dot{\chi}_{j,{a}\setminus i} W_{j} \right). \end{array} $$

From the proof of Proposition 1, recall that W(A ∪ B) = W(A) + W(B) almost surely for disjoint Borel sets A and B of [0, ∞]^d ∖{∞} bounded away from ∞; moreover, for such A and B, the variables W(A) and W(B) are uncorrelated. Since R_a∖i is the disjoint union of R_a and R_a∖i ∖ R_a, we have therefore $W_{{a}\setminus i}^{\cap } = W_{{a}}^{\cap } + W(R_{{a}\setminus i} \setminus R_{{a}})$ almost surely. In addition, ${\sum }_{i \in {a}} \chi _{{a}\setminus i} = \mu ({\Delta }_{{a}}) + (S-1)\chi _{{a}}$ by Eq. 11 applied to ν = μ. As a consequence,

$$H_{{a}} = \{ \mu({\Delta}_{{a}}) - \chi_{{a}} \} W_{{a}}^{\cap} - \chi_{{a}} \sum\limits_{j \in{a}} W(R_{{a}\setminus j} \setminus R_{{a}}) + \sum\limits_{j\in{a}} K_{{a}, j} W_{j} $$

where

$$K_{{a},j} = \chi_{{a}}\left( \sum\limits_{i\in{a}\setminus j}\dot{\chi}_{j,{a}\setminus i}\right) - \left( \sum\limits_{i\in{a}}\chi_{{a}\setminus i}\right)\dot{\chi}_{j,{a}}, \qquad j \in {a}. $$

The S + 1 variables $W_{{a}}^{\cap } = W(R_{{a}})$ and W(R_a∖j ∖ R_a), j ∈ a, are all uncorrelated, since they involve evaluating W at disjoint sets; W_j = W(R_{j}) is uncorrelated with W(R_a∖j ∖ R_a), for the same reason. Moreover, ${\mathbb {E}}[ W_{{a}}^{\cap } W_{j} ] = {\Lambda } (R_{{a}} \cap R_{\{j\}}) = {\Lambda } (R_{{a}}) = \chi _{{a}}$ and similarly ${\mathbb {E}}[ W(R_{{a}\setminus i} \setminus R_{{a}}) W_{j} ] = {\Lambda } (R_{{a}\setminus i} \setminus R_{{a}}) = \chi _{{a} \setminus i} - \chi _{{a}}$ if i, j ∈ a and i ≠ j. Hence

$$\begin{array}{@{}rcl@{}} {\mathbb{V}\text{ar}}(H_{{a}}) &=& \{\mu({\Delta}_{{a}}) - \chi_{{a}}\}^{2} \chi_{{a}} + \chi_{{a}}^{2} \sum\limits_{j \in {a}} (\chi_{{a} \setminus j} - \chi_{{a}}) + \sum\limits_{i,j \in {a}} K_{{a},i} K_{{a},j} \chi_{\{i,j\}} \\ &&+ \{\mu({\Delta}_{{a}}) - \chi_{{a}}\} \chi_{{a}}\sum\limits_{j\in{a}}K_{{a},j} - \chi_{{a}} \sum\limits_{j\in{a}}K_{{a},j} \sum\limits_{i\in{a}\setminus j} (\chi_{{a}\setminus i} - \chi_{{a}}). \end{array} $$

As ${\sum }_{j \in {a}} (\chi _{{a} \setminus j} - \chi _{{a}}) = \mu ({\Delta }_{{a}})- \chi _{{a}}$ and ${\sum }_{i \in {a} \setminus j}(\chi _{{a} \setminus i} - \chi _{{a}}) = \mu ({\Delta }_{{a}}) - \chi _{{a},j}$, we obtain

$$\begin{array}{@{}rcl@{}} {\mathbb{V}\text{ar}}(H_{{a}}) &=& \{\mu({\Delta}_{{a}}) - \chi_{{a}}\}\chi_{{a}} \left\{ \mu({\Delta}_{{a}}) + \sum\limits_{j\in{a}}K_{{a},j} \right\} + \sum\limits_{i,j\in{a}}K_{{a},i}K_{{a},j} \chi_{\{i,j\}}\\ &&-\chi_{{a}} \sum\limits_{j \in{a}}K_{{a},j}\{\mu({\Delta}_{{a}}) - \chi_{{a}\setminus j} \}. \end{array} $$

(30)

Recall κ_a(x) in Eq. 15. We have

$$\begin{array}{@{}rcl@{}} \frac{\partial}{\partial x_{j}} \left( \frac{1}{\kappa_{{a}}(\boldsymbol{x})} \right)_{\boldsymbol{x} = \boldsymbol{1}_{{a}}} &=& \frac{\partial}{\partial x_{j}} \left( \frac{{\sum}_{i \in {a}} r_{{a} \setminus i}(\boldsymbol{x}_{{a} \setminus i})}{r_{{a}}(\boldsymbol{x})} \right) \\ &=& \chi_{{a}}^{-2} \left( \chi_{{a}} \sum\limits_{i \in {a} \setminus j} \dot{\chi}_{j, {a} \setminus i} - \dot{\chi}_{j, {a}} \sum\limits_{i \in {a}} \chi_{{a} \setminus i} \right) = \chi_{{a}}^{-2} K_{{a}, j}. \end{array} $$

It follows that $\dot {\kappa }_{j,{a}} = - \chi _{{a}}^{-2} K_{{a}, j} / (1/\kappa _{{a}})^{2} = - K_{{a},j} / \mu ({\Delta }_{{a}})^{2}$. By Eq. 30, we find that $\sigma _{\kappa ,{a}}^{2} = \mu ({\Delta }_{{a}})^{-4} {\mathbb {V}\text {ar}}(H_{{a}} )$ is equal to the right-hand side of Eq. 17. □

Proof of Proposition 3

We only need to prove that $\widehat {\sigma }^{2}_{\kappa ,{a}} = \sigma _{\kappa , {a}}^{2} + \mathrm {o}_{{\mathbb {P}}}(1)$ as n →∞. In view of the expressions (17) and (19) for $\sigma _{\kappa , {a}}^{2}$ and $\widehat {\sigma }_{\kappa ,{a}}$, it is enough to show that $\dot {\kappa }_{j,{a}, n} = \dot {\kappa }_{{a},j} + \mathrm {o}_{{\mathbb {P}}}(1)$, with $\dot {\kappa }_{j,{a}, n}$ in Eq. 18; indeed, Corollary 1 already gives consistency of $\widehat {\mu }({\Delta }_{{a}})$ and $\widehat {\chi }_{b}$. Now since $2^{-1} k^{1/4} \{ \kappa _{{a}}(\boldsymbol {1}_{{a}} + k^{-1/4}\boldsymbol {e}_{j}) - \kappa _{{a}}(\boldsymbol {1}_{{a}} - k^{-1/4}\boldsymbol {e}_{j} ) \} \to \dot {\kappa }_{{a},j}$ as n →∞, a sufficient condition is that for some ε > 0,

$$ \sup_{[1-\varepsilon,2+\varepsilon]^{{a}}} k^{1/4}\big|\widehat \kappa_{{a}}(\boldsymbol{x}) - \kappa_{{a}}(\boldsymbol{x})\big| = \mathrm{o}_{{\mathbb{P}}}(1), \qquad n \to \infty. $$

(31)

In turn, Eq. 31 follows from weak convergence of $k^{1/2}(\widehat {\kappa }_{{a}} - \kappa _{{a}} )$ as n →∞ in the space ℓ^∞([1 − ε, 1 + ε]^a). In light of the expressions of $\widehat {\kappa }_{{a}}$ and κ_a in terms of the (empirical) joint tail dependence functions $\widehat {r}_{b}$ and r_b, respectively, weak convergence of $k^{1/2}(\widehat {\kappa }_{{a}} - \kappa _{{a}} )$ follows from Proposition 1 and the functional delta method (van der Vaart 1998, Theorem 20.8). The calculations are similar to the ones for the Euclidean case in the proof of Proposition 2; an extra point to be noted is that if a is such that μ(Δ_a) > 0, then the denominator in the definition of κ_a(x) in Eq. 31 is positive for all x in a neighbourhood of 1_a. □

Proof of Proposition 4

Proposition 1 implies, as n →∞, the weak convergence

$$\left( \sqrt k\{ \widehat r_{{a}}(\boldsymbol{2}_{{a}}) - r_{{a}}(\boldsymbol{2}_{{a}} \}, \sqrt k\{ \widehat r_{{a}}(\boldsymbol{1}_{{a}}) - r_{{a}}(\boldsymbol{1}_{{a}} \} \right) \rightsquigarrow \left( Z_{{a}}(\boldsymbol{2}_{{a}}), Z_{{a}}(\boldsymbol{1}_{{a}}) \right). $$

Now $\widehat \eta _{{a}}^{P} = g(\widehat r_{{a}}(\boldsymbol {2}_{{a}}), \widehat r_{{a}} (\boldsymbol {1}_{{a}}))$ and η_a = 1 = g(r_a(2_a), r_a(1_a)) = g(2χ_a, χ_a), with g(x, y) = log(2)/ log(x/y); note that the function r_a is homogeneous. The gradient of g is ∇g(x, y) = log(2)(log(x/y))^− 2(−x^− 1, y^− 1), so that the delta method gives

$$\begin{array}{@{}rcl@{}} &&\sqrt k (\widehat \eta^{P} - 1) \rightsquigarrow \left\langle \nabla g(2\chi_{{a}}, \chi_{{a}}) , \left( Z_{{a}}(\boldsymbol{2}_{{a}}), Z_{{a}}(\boldsymbol{1}_{{a}}) \right) \right\rangle\\ && = \frac{1}{\chi_{{a}} \log 2 } \left\langle (-1/2, 1 ) , \left( Z_{{a}}(\boldsymbol{2}_{{a}}), Z_{{a}}(\boldsymbol{1}_{{a}})\right) \right\rangle\\ && = \frac{-1}{2 \chi_{{a}} \log 2 } \{ Z_{{a}}(\boldsymbol{2}_{{a}}) - 2 Z_{{a}}(\boldsymbol{1}_{{a}}) \}. \end{array} $$

The first part of the assertion follows. As for the variance,

$$\begin{array}{@{}rcl@{}} {\mathbb{V}\text{ar}}(Z_{{a}}(\boldsymbol{2}_{{a}}) - 2Z_{{a}}(\boldsymbol{1}_{{a}}) ) &=& {\mathbb{V}\text{ar}}(Z_{{a}}(\boldsymbol{2}_{{a}}) ) + 4{\mathbb{V}\text{ar}}(Z_{{a}}(\boldsymbol{1}_{{a}}) ) - 4 {\mathbb{C}\text{ov}}(Z_{{a}}(\boldsymbol{2}_{{a}}), Z_{{a}}(\boldsymbol{1}_{{a}})), \end{array} $$

The function r_a is homogeneous of order 1, so that ∂_jr_a is constant along rays, that is, the function 0 < t↦∂_jr_a(tx) is constant. Moreover, the measure Λ is homogeneous of order 1 too. In view of Eqs. 9 and 10, it follows that ${\mathbb {V}\text {ar}}(Z_{{a}}(t \boldsymbol {x}) ) = t {\mathbb {V}\text {ar}}(Z_{{a}}(\boldsymbol {x} ) )$ for t > 0; in particular ${\mathbb {V}\text {ar}}(Z_{{a}}(\boldsymbol {2}_{{a}}) = 2 {\mathbb {V}\text {ar}}(Z_{{a}}(\boldsymbol {1}_{{a}} )$. Further, $\chi _{{a}} = (\mathrm {d} r_{{a}}(t, \ldots , t) / \mathrm {d}t)_{t = 1} = {\sum }_{j \in {a}} \dot {\chi }_{j,{a}}$ and thus

$$\begin{array}{@{}rcl@{}} {\mathbb{V}\text{ar}}(Z_{{a}}(\boldsymbol{1}_{{a}})) & =& \chi_{{a}} - 2\sum\limits_{j\in{a}}\dot{\chi}_{j,{a}} \chi_{{a}} + \sum\limits_{j\in{a}} \sum\limits_{j^{\prime}\in{a}}\dot{\chi}_{j,{a}}\dot{\chi}_{j^{\prime},{a}} \chi_{\{j,j^{\prime}\}} \\ & =& \chi_{{a}} - 2 \chi_{{a}}^{2} + \sum\limits_{j\in{a}} \sum\limits_{j^{\prime}\in{a}}\dot{\chi}_{j,{a}}\dot{\chi}_{j^{\prime},{a}} \chi_{\{j,j^{\prime}\}}. \end{array} $$

The covariance term is

$$\begin{array}{@{}rcl@{}} {\mathbb{C}\text{ov}}(Z_{{a}}(\boldsymbol{2}_{{a}}), Z_{{a}}(\boldsymbol{1}_{{a}})) &=&\chi_{{a}} - \sum\limits_{j\in{a}} \dot{\chi}_{j,{a}} \chi_{{a}} - \sum\limits_{j\in{a}}\dot{\chi}_{j,{a}} r_{{a}}(\boldsymbol{2}_{{a}} \wedge \boldsymbol{\iota}_{j})\\ &&+ \sum\limits_{j\in{a}} \sum\limits_{j^{\prime}\in{a}} \dot{\chi}_{j,{a}}\dot{\chi}_{j^{\prime},{a}}r_{\{j,j^{\prime}\}}(2,1), \end{array} $$

(32)

with 2_a ∧ι_j as explained in the statement of the proposition. Since ${\sum }_{j\in {a}} \dot {\chi }_{j,{a}} = \chi _{{a}}$, we can simplify and find

$$\begin{array}{@{}rcl@{}} {\mathbb{V}\text{ar}}(Z_{{a}}(\boldsymbol{2}_{{a}}) - 2Z_{{a}}(\boldsymbol{1}_{{a}}) ) &=& 6 {\mathbb{V}\text{ar}}(Z_{{a}}(\boldsymbol{1}_{{a}}) ) - 4 {\mathbb{C}\text{ov}}(Z_{{a}}(\boldsymbol{2}_{{a}}), Z_{{a}}(\boldsymbol{1}_{{a}})) \\ &=& 2 \chi_{{a}} - 8 \chi_{{a}}^{2} + 4\sum\limits_{j\in{a}} \dot{\chi}_{j,{a}} r_{{a}}(\boldsymbol{2}_{{a}} \wedge \boldsymbol{\iota}_{j})\\ &&+ \sum\limits_{j\in{a}} \sum\limits_{j^{\prime}\in{a}} \dot{\chi}_{j,{a}}\dot{\chi}_{j^{\prime},{a}} \left[ 6 \chi_{\{j, j^{\prime}\}} - 4 r_{\{j, j^{\prime} \}}(2,1)\right]. \end{array} $$

Divide the right-hand side by (2χ_a log 2)² to obtain Eq. 22. □

Proof of Proposition 6

To alleviate notations, ∅ ≠ a ⊂{1,…, d} is fixed and the subscript a is omitted throughout the proof. Introduce the tail empirical process $Q_{n}(t) = \widehat T_{(n - \lfloor kt \rfloor )}$ for 0 < t < n/k. The key is to represent the Hill estimator as a statistical tail functional (Drees (1998a), Example 3.1) of Q_n, i.e., $\widehat \eta ^{H} = {\Theta }(Q_{n})$, where Θ is the map defined for any measurable function $z: (0,1]\to \mathbb {R}$ as ${\Theta }(z) = {{\int }_{0}^{1}} \log ^{+} \{ z(t) / z(1) \} \mathrm {d} t$ when the integral is finite and Θ(z) = 0 otherwise. Let z_η : t ∈ (0, 1]↦t^−η denote the quantile function of a standard Pareto distribution with index 1/η; it holds that Θ(z_η) = η. The map Θ is scale invariant, i.e., Θ(tz) = Θ(z), t > 0.

The proof consists of three steps:

• 1. Introduce a function space D_{η, h} making it possible to control Q_n(t) and z_η(t) as t → 0. In this space and up to rescaling, Q_n − z_η converges weakly to a Gaussian process.

• 2. Show that the map Θ is Hadamard differentiable at z_η tangentially to some well chosen subspace of D_{η, h}.

• 3. Apply the functional delta method to show that η^H = Θ(Q_n) is asymptotically normal and compute its asymptotic variance via the Hadamard derivative of Θ.

• Step 1. Let ε > 0 and h(t) = t^{1/2 + ε}, t ∈ [0, 1]. Then $h\in \mathcal {H}$, where

  $$\mathcal{H} = \{z: [0,1]\to \mathbb{R} \mid z \text{ continuous, } \lim_{t\to 0} z(t) t^{-1/2} (\log\log(1/t))^{1/2} = 0 \}.$$

  Introduce the function space

  $$D_{\eta, h} = \{z: [0,1]\to \mathbb{R} \mid \lim_{t\to 0} t^{\eta} h(t) z(t) = 0 ; t\mapsto t^{\eta}h(t) z(t) \in D[0,1] \}, $$

  where D[0, 1] is the space of càdlàg functions. Notice that z_η ∈ D_{η, h}. Equip D_{η, h} with the seminorm ∥z∥_{η, h} = sup_{t∈(0, 1]}|t^ηh(t)z(t)|. Let m = ⌈nq^←(k/n)⌉, with ⌈⋅⌉ the ceil function, so that k/m → χ; for self-consistency of the present paper, the roles of k and m are reversed compared to the notation in Draisma et al. (2004). From Draisma et al. (2004, Lemma 6.2), we have, for all t₀ > 0, in the space D_{η, h}, the weak convergence

  $$ \sqrt{k} \left( \frac{m}{n}Q_{n} - z_{\eta} \right) \rightsquigarrow \left( \eta t^{-(\eta + 1)} \bar W(t)\right)_{t \in [0,t_{0}]} $$

  (33)

  where $\bar W (t) = \tilde W(\boldsymbol {t}_{{a}})$, and $\tilde W$ is defined as in the statement of Proposition 6. Indeed, the process $\bar W$ in the statement from Draisma et al. (2004, Lemmata 6.1 and 6.2) has same distribution as W₁(t_a) in the case χ = 0; recall that our χ is denoted by l in Draisma et al. (2004). Put U_{i, j} = 1 − F_j(X_{i, j}), and let U_{(1), j} ≤… ≤ U_{(d), j} be the order statistics of U_{1, j},…, U_{n, j}. In the case χ > 0, $\bar W$ equals in distribution W_dra(t_a) where W_dra appears in Lemma 6.1 in the cited reference as the limit in distribution (for a = {1, 2}), for x ∈ E_a, of

  

  From Proposition 1 and Slutsky’s Lemma, we have ${\Delta }_{n,k,m} \rightsquigarrow \chi ^{-1/2} Z_{{a}}$ in ℓ^∞([0, 1]^a). Therefore, W_dra = χ^− 1/2Z_a, as claimed.

• Step 2. The right-hand side of Eq. 33 belongs to $\mathcal {C}_{h,\eta } = \{ z \in D_{\eta ,h} \mid \text {\textit {z} is continuous}\}$. To apply the functional delta-method (van der Vaart 1998, Theorem 20.8), we must verify that the restriction of Θ to $\bar D_{\eta , h}$ is Hadamard-differentiable tangentially to $\mathcal {C}_{\eta ,h}$, with derivative Θ^′, where $\bar D_{\eta , h}$ is a subspace of D_{η, h} such that ${\mathbb {P}}(Q_{n} \in \bar D_{\eta ,h})\to 1$ as n →∞; see the remark following Condition 3 in Drees (1998a). Then it will follow from the scale invariance of Θ, the identities ${\Theta }(Q_{n}) = \widehat \eta ^{H}$ and Θ(z_η) = η, and the weak convergence in Eq. 33 that

  $$ \sqrt{k} \left( \widehat\eta^{H} - \eta \right) = \sqrt{k} \left( {\Theta}\left( \frac{m}{n}Q_{n}\right) - {\Theta}(z_{\eta}) \right) \rightsquigarrow {\Theta}^{\prime}\left[\left( \eta t^{-(\eta + 1)} \bar W(t)\right)_{t \in [0,1]}\right] $$

  (34)

  as n →∞. From Drees (1998a, Example 3.1), the restriction of Θ to $\bar D_{\eta ,h}$, the subset of functions on D_{η, h} which are positive and non increasing, is indeed Hadamard differentiable; letting ν denote the measure dν(t) = t^ηdt + dε₁(t), with ε₁ a point mass at 1, the derivative is

  $${\Theta}^{\prime}(z) = {{\int}_{0}^{1}} t^{\eta} z(t) \mathrm{d} t - y(1) = {\int}_{[0,1]} z(t) \mathrm{d} \nu (t). $$

• Step 3. The weak limit in Eq. 34 is thus equal to ${\int }_{[0,1]} \eta t^{-(\eta + 1)} \bar W(t) \mathrm {d} \nu (t)$. From Shorack and Wellner (2009, Proposition 2.2.1), the latter random variable is centered Gaussian with variance

  $$\sigma^{2} = \int\!\!\! {\int}_{[0,1]^{2}} \eta^{2} (st)^{-(\eta + 1)} {\mathbb{C}\text{ov}}(\bar W(s), \bar W(t)) \mathrm{d} \nu(s) \mathrm{d}\nu(t). $$

  By definition of ν and by symmetry of the covariance,

  $$\begin{array}{@{}rcl@{}} \sigma^{2} / \eta^{2} &=& 2 \underbrace{{\int}_{s = 0}^{1}{\int}_{t = 0}^{s} (st)^{-1} {\mathbb{C}\text{ov}}(\bar W(s), \bar W(t)) \mathrm{d} t \mathrm{d} s}_{A}\\ &&- 2 \underbrace{{\int}_{s = 0}^{1} {\mathbb{C}\text{ov}}(\bar W(s), \bar W(1)) s^{-1} \mathrm{d} s}_{B} + {\mathbb{V}\text{ar}}(\bar W(1)). \end{array} $$

  For any s ∈ (0, 1),

  $$\begin{array}{@{}rcl@{}} {\int}_{t = 0}^{s} {\mathbb{C}\text{ov}}(\bar W(s), \bar W(t))(st)^{-1} \mathrm{d} t & =&{\int}_{u = 0}^{1} {\mathbb{C}\text{ov}}(\bar W(s), \bar W(us))(su)^{-1} \mathrm{d} u \\ &=& {\int}_{u = 0}^{1} {\mathbb{C}\text{ov}}(\bar W(1), \bar W(u))(u)^{-1} \mathrm{d} u = B. \end{array} $$

  The penultimate equality follows from ${\mathbb {C}\text {ov}}(\bar W(\lambda s),\bar W(\lambda t)) = \lambda {\mathbb {C}\text {ov}}(\bar W(s),\bar W(t))$ for λ > 0 and s, t ∈ (0, 1]. Therefore A = B and $\sigma ^{2} = \eta ^{2} {\mathbb {V}\text {ar}}(\bar W(1))$, as required.

□

Appendix B: CLEF algorithm and variants

The CLEF algorithm is described at length in Chiapino and Sabourin (2016). For completeness, its pseudo-code is provided below. The underlying idea is to iteratively construct pairs, triplets, quadruplets… of features that are declared ‘dependent’ whenever $\widehat \kappa _{{a}} \ge C$ for some user-defined tolerance level C > 0. Varying this criterion produces three variants of the original algorithm, namely CLEF-Asymptotic, CLEF-Peng, and CLEF-Hill. The pruning stage of the algorithm is the same for all three variants.