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Stable tail dependence functions – some basic properties

Stable tail dependence functions – some basic properties 1IntroductionA multivariate extreme value (MEV) distribution (in a standardised form) is given by a distribution function (“d.f.”) FFon R+d{{\mathbb{R}}}_{+}^{d}with the decisive property (F(tx))t=F(x)∀x∈R+d,∀t>0,{\left(F\left(tx))}^{t}=F\left(x)\hspace{1.0em}\forall x\in {{\mathbb{R}}}_{+}^{d},\hspace{1em}\forall t\gt 0,and with standard one-dimensional Fréchet margins, defined by the d.f. exp−1u\exp \left(-\frac{1}{u}\right)for u>0u\gt 0. The d.f. FFis in a one-to-one correspondence with its associated stable tail dependence function (“STDF”), defined by ℓ(x)≔−logF1x,x∈R+d,\ell \left(x):= -\log F\left(\frac{1}{x}\right),\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d},where 1x≔1x1,1x2,…\frac{1}{x}:= \left(\frac{1}{{x}_{1}},\frac{1}{{x}_{2}},\ldots \right), and these STDFs allow an intrinsic characterisation: ℓ:R+d→R\ell :{{\mathbb{R}}}_{+}^{d}\to {\mathbb{R}}is a STDF iff ℓ\ell is homogeneous (ℓ(tx)=t⋅ℓ(x)∀t,∀x\ell \left(tx)=t\cdot \ell \left(x)\hspace{1em}\forall t,\forall x), normalised (ℓ(ei)=1∀\ell \left({e}_{i})=1\hspace{1em}\forall unit vector ei{e}_{i}), and “fully dd-alternating” (to be explained later on), cf. [6], Theorem 6.The marginals of FFare given by Fα(xα)≔F(xα,∞αc)=exp−ℓ1xα,0αc{F}_{\alpha }\left({x}_{\alpha }):= F\left({x}_{\alpha },{\infty }_{{\alpha }^{c}})=\exp \left[-\ell \left(\frac{1}{{x}_{\alpha }},{{\bf{0}}}_{{\alpha }^{c}}\right)\right]for ∅≠α⊆{1,…,d},xα∈R+α\varnothing \ne \alpha \subseteq \left\{1,\ldots ,d\right\},{x}_{\alpha }\in {{\mathbb{R}}}_{+}^{\alpha }. Fα{F}_{\alpha }is again a MEV distribution with STDF ℓα≔ℓ∣R+α{\ell }_{\alpha }:= \ell | {{\mathbb{R}}}_{+}^{\alpha }. If X=(X1,…,Xd)X=\left({X}_{1},\ldots ,{X}_{d})has the d.f. FF, the subvector Xα≔(Xi,i∈α){X}_{\alpha }:= \left({X}_{i},i\in \alpha )has d.f. Fα{F}_{\alpha }.Two main subjects will be treated in this article. The first one is about the so-called extremal coefficients of a (dd-variate) STDF ℓ\ell , defined by ℓ(α)≔ℓ(1α),α⊆[d]\ell \left(\alpha ):= \ell \left({{\bf{1}}}_{\alpha }),\hspace{1.0em}\alpha \subseteq \left[d](slightly abusing notation). Although ℓ\ell is plainly not determined by its restriction to {0,1}d{\left\{0,1\right\}}^{d}, these coefficients contain important information, especially with respect to the independence of subvectors (Theorem 3).The other main theme addressed is about logistic, negative logistic and nested logistic STDFs. A certain functional equation (Theorem 5) turns out to be the key for several characterisations of “Archimedean type.” The well known sufficient conditions for “composebility” of logistic STDFs are shown to be necessary as well (Theorem 9) – meaning that the composed function is again a STDF.Except Theorem 1, all the other theorems in this article are new to the best of our knowledge. A recommendable treatment of STDFs is presented in chapter 8 of [1].Notations:R+≔[0,∞[{{\mathbb{R}}}_{+}:= {[}0,\infty {[}, N≔{1,2,3,…}{\mathbb{N}}:= \left\{1,2,3,\ldots \right\}, N0≔{0,1,2,…}{{\mathbb{N}}}_{0}:= \left\{0,1,2,\ldots \right\}, R¯≔[−∞,∞]\overline{{\mathbb{R}}}:= \left[-\infty ,\infty ],1x≔1x1,1x2,…\frac{1}{x}:= \left(\frac{1}{{x}_{1}},\frac{1}{{x}_{2}},\ldots \right)with 10≔∞,1∞≔0\frac{1}{0}:= \infty ,\frac{1}{\infty }:= 0[d]≔{1,…,d}\left[d]:= \left\{1,\ldots ,d\right\}, −α≔αc=[d]⧹α-\alpha := {\alpha }^{c}=\left[d]\setminus \alpha for a⊆[d]a\subseteq \left[d], 1d≔(1,…,1)∈Nd{{\bf{1}}}_{d}:= \left(1,\ldots ,1)\in {{\mathbb{N}}}^{d}, e1,…,ed{e}_{1},\ldots ,{e}_{d}are the usual unit vectors in Rd{{\mathbb{R}}}^{d}, 1α≔∑i∈αei{{\bf{1}}}_{\alpha }:= \sum _{i\in \alpha }{e}_{i}(f×g)(x,y)≔(f(x),g(y))(f\times g)\left(x,y):= (f\left(x),g(y))for mappings f,gf,gM+(X){M}_{+}\left(X)is the set of Radon measures on a locally compact space XXd.f. = distribution function.2Fully d-alternating functionsTo define this notion, which is of particular importance in this article, we introduce a special notation for multivariate real-valued functions. Let A1,…,Ad{A}_{1},\ldots ,{A}_{d}be non-empty sets, A≔A1×⋯×AdA:= {A}_{1}\times \cdots \times {A}_{d}, and f:A→Rf:A\to {\mathbb{R}}. First, for x∈Ax\in Aand ∅≠u⊆[d]\varnothing \ne u\subseteq \left[d], we put xu≔(xi)i∈u{x}_{u}:= {\left({x}_{i})}_{i\in u}, −u≔[d]⧹u-u:= \left[d]\setminus u, and so for x,z∈Ax,z\in A(zu,x−u)≔zi,i∈uxi,i∈−u,\left({z}_{u},{x}_{-u}):= \left\{\begin{array}{l}{z}_{i},i\in u\hspace{1.0em}\\ {x}_{i},i\in -u,\hspace{1.0em}\end{array}\right.i.e., another element of AA, being xxfor u=∅u=\varnothing and zzfor u=[d]u=\left[d]. Also, Au≔∏i∈uAi{A}_{u}:= {\prod }_{i\in u}{A}_{i}for u≠∅u\ne \varnothing . We then define Dzxf≔∑u⊆[d](−1)∣u∣f(zu,x−u)=f(x)∓⋯+(−1)df(z).{D}_{z}^{x}f:= \sum _{u\subseteq \left[d]}{\left(-1)}^{| u| }f\left({z}_{u},{x}_{-u})=f\left(x)\hspace{0.33em}\mp \cdots +\hspace{0.33em}{\left(-1)}^{d}f\left(z).Note that Dzxf=(−1)dDxzf{D}_{z}^{x}f={\left(-1)}^{d}{D}_{x}^{z}f. For ∅≠u⊊[d]\varnothing \ne u\hspace{0.33em}\subsetneq \hspace{0.33em}\left[d]and y−u∈A−u{y}_{-u}\in {A}_{-u}, we define a “partial version” of ffwith fixed values in the variables i∈−ui\in -uby f(⋅,y−u)(xu)≔f(xu,y−u),xu∈Au.f\left(\cdot ,{y}_{-u})\left({x}_{u}):= f\left({x}_{u},{y}_{-u}),\hspace{1.0em}{x}_{u}\in {A}_{u}.(For u=[d]u=\left[d], this would be ff, and for u=∅u=\varnothing , the constant f(y[d])f({y}_{\left[d]}).)There is a two-step procedure to determine Dzxf{D}_{z}^{x}fwhich will be needed later on:Lemma 1Let f:A→Rf:A\to {\mathbb{R}}, ∅≠v⊊[d]\varnothing \ne v\hspace{0.33em}\subsetneq \hspace{0.33em}\left[d], x,z∈Ax,z\in A, and define g:A−v→Rg:{A}_{-v}\to {\mathbb{R}}byg(y−v)≔Dzvxvf(⋅,y−v),y−v∈A−v.g({y}_{-v}):= {D}_{{z}_{v}}^{{x}_{v}}f\left(\cdot ,{y}_{-v}),\hspace{1.0em}{y}_{-v}\in {A}_{-v}.ThenDzxf=Dz−vx−vg.{D}_{z}^{x}f={D}_{{z}_{-v}}^{{x}_{-v}}g.ProofDz−vx−vg=∑w⊆−v(−1)∣w∣g(zw,x(−v)⧹w)=∑w⊆−v(−1)∣w∣∑u⊆v(−1)∣u∣f(zu,xv⧹u,zw,x(−v)⧹w)(notingu∩w=∅here, puttingα≔u∪w)=∑α⊆[d](−1)∣α∣f(zα,x−α)=Dzxf.□\hspace{12em}\begin{array}{rcl}{D}_{{z}_{-v}}^{{x}_{-v}}g& =& \displaystyle \sum _{w\subseteq -v}{\left(-1)}^{| w| }g\left({z}_{w},{x}_{\left(-v)\setminus w})\\ & =& \displaystyle \sum _{w\subseteq -v}{\left(-1)}^{| w| }\displaystyle \sum _{u\subseteq v}{\left(-1)}^{| u| }f\left({z}_{u},{x}_{v\setminus u},{z}_{w},{x}_{\left(-v)\setminus w})\\ & & \left(\hspace{0.1em}\text{noting}\hspace{0.1em}\hspace{0.33em}u\cap w=\varnothing \hspace{0.33em}\hspace{0.1em}\text{here, putting}\hspace{0.1em}\hspace{0.33em}\alpha := u\cup w)\\ & =& \displaystyle \sum _{\alpha \subseteq \left[d]}{\left(-1)}^{| \alpha | }f\left({z}_{\alpha },{x}_{-\alpha })\\ & =& {D}_{z}^{x}f.\hspace{28em}\square \end{array}DefinitionLet A1,…,Ad⊆R¯{A}_{1},\ldots ,{A}_{d}\subseteq \overline{{\mathbb{R}}}be non-empty, A≔A1×⋯×AdA:= {A}_{1}\times \cdots \times {A}_{d}. Then f:A→Rf:A\to {\mathbb{R}}is fully d-alternating (in symbols “1d-↕{{\bf{1}}}_{d}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow ”) iff Dzxf≤0{D}_{z}^{x}f\le 0for x≤zx\le z(both in AA), and if also Dzvxvf(⋅,y−v)≤0{D}_{{z}_{v}}^{{x}_{v}}f\left(\cdot ,{y}_{-v})\le 0for each ∅≠v⊊[d]\varnothing \ne v\hspace{0.33em}\subsetneq \hspace{0.33em}\left[d]and xv≤zv{x}_{v}\le {z}_{v}(both in Av{A}_{v}), and for all y−v∈A−v{y}_{-v}\in {A}_{-v}.This property is specific for co-survival functions, i.e., f(x)≔P(X≱x)f\left(x):= P\left(X\ge &#x0338;x), e.g., for XXuniform on [0,1]d{\left[0,1]}^{d}f(x)=∑xi−∑i<jxixj+∑i<j<kxixjxk∓⋯,f\left(x)=\sum {x}_{i}-\sum _{i\lt j}{x}_{i}{x}_{j}+\sum _{i\lt j\lt k}{x}_{i}{x}_{j}{x}_{k}\mp \cdots \hspace{0.33em},but it is of special importance also for some infinite measures, as we will see shortly.Remark 1There is a more general notion of n{\bf{n}}-alternating (“n-↕{\bf{n}}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow ”) functions, with n∈Nd{\bf{n}}\in {{\mathbb{N}}}^{d}, cf. [7], describing monotonicity conditions of higher orders, not needed in this work.For A1=⋯=Ad=R+{A}_{1}=\cdots ={A}_{d}={{\mathbb{R}}}_{+}, i.e., A=R+dA={{\mathbb{R}}}_{+}^{d}, a function ℓ:A→R\ell :A\to {\mathbb{R}}is a STDF iff ℓ\ell is homogeneous (i.e., ℓ(tx)=t⋅ℓ(x)∀t≥0,∀x∈R+d\ell \left(tx)=t\cdot \ell \left(x)\hspace{1em}\forall t\ge 0,\hspace{1em}\forall x\in {{\mathbb{R}}}_{+}^{d}), 1d-↕{{\bf{1}}}_{d}\hspace{-0.33em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.3em}\updownarrow and normalised, i.e., ℓ(ei)=1\ell \left({e}_{i})=1for each unit vector. Disregarding normalisation, we consider K≔{f:R+d→R+∣fis1d-↕and homogeneous,f(1d)=1}.K:= \{f:{{\mathbb{R}}}_{+}^{d}\to {{\mathbb{R}}}_{+}| f\hspace{0.33em}\hspace{0.1em}\text{is}\hspace{0.1em}\hspace{0.33em}{{\bf{1}}}_{d}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow \hspace{0.33em}\hspace{0.1em}\text{and homogeneous}\hspace{0.1em},\hspace{0.33em}f\left({{\bf{1}}}_{d})=1\}.This set, obviously compact and convex, was shown in ref. [6] to be a so-called Bauer simplex (i.e., a compact convex subset of some locally convex Hausdorff space, for which the extreme boundary is closed, and for which the integral representation given by the Krein-Milman theorem is unique), with extreme boundary ex(K)={x↦maxi≤d(xiwi)∣w∈Cd},{\rm{ex}}\left(K)=\left\{\phantom{\rule[-1.25em]{}{0ex}}x\mapsto \mathop{\max }\limits_{i\le d}\left({x}_{i}{w}_{i})| w\in {C}_{d}\right\},where Cd≔{w∈[0,1]d∣maxi≤dwi=1}{C}_{d}:= \left\{w\in {\left[0,1]}^{d}| {\max }_{i\le d}{w}_{i}=1\right\}, and for each homogeneous 1d-↕{{\bf{1}}}_{d}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow function ff(≢0\not\equiv 0), we have the unique integral representation f(x)=f(1d)⋅∫maxi≤d(xiwi)dν(w)f\left(x)=f\left({{\bf{1}}}_{d})\cdot \int \mathop{\max }\limits_{i\le d}\left({x}_{i}{w}_{i}){\rm{d}}\nu \left(w)with ν\nu a probability measure on Cd{C}_{d}.The function ffis then the so-called co-survival function of a homogeneous Radon measure μ\mu on the locally compact space Zd≔[0,∞]d⧹{∞d}{Z}_{d}:= {\left[0,\infty ]}^{d}\setminus \left\{{\infty }_{d}\right\}, i.e., f(x)=μ([x,∞]c)≕μˇ(x)f\left(x)=\mu \left({\left[x,\infty ]}^{c})\hspace{0.33em}=: \hspace{0.33em}\check{\mu }\left(x)(which is finite by the definition of a Radon measure).3Properties depending on the extremal coefficientsLet ℓ\ell be a dd-variate STDF. Its restriction ℓ∣{0,1}d\ell | {\left\{0,1\right\}}^{d}gives the so-called extremal coefficients ℓ(α)≔ℓ(1α)\ell \left(\alpha ):= \ell \left({{\bf{1}}}_{\alpha })for α⊆[d]\alpha \subseteq \left[d](hence, ℓ([d])=ℓ(1d)\ell \left(\left[d])=\ell \left({{\bf{1}}}_{d})). From the integral representation, ℓ(x)=ℓ(1d)⋅∫Cdmaxi≤d(xiwi)dν(w),\ell \left(x)=\ell \left({{\bf{1}}}_{d})\cdot \mathop{\int }\limits_{{C}_{d}}\mathop{\max }\limits_{i\le d}\left({x}_{i}{w}_{i}){\rm{d}}\nu \left(w),we obtain ℓ(α)=ℓ(1d)⋅∫maxi∈αwidν(w),∅≠α⊆[d].\ell \left(\alpha )=\ell \left({{\bf{1}}}_{d})\cdot \int \mathop{\max }\limits_{i\in \alpha }{w}_{i}{\rm{d}}\nu \left(w),\hspace{1.0em}\varnothing \ne \alpha \subseteq \left[d].Clearly, the restriction of ℓ\ell to {0,1}d{\left\{0,1\right\}}^{d}does in general not determine ℓ\ell , with the following exception, known since a long time, and due to Takahashi [9,10]. Note that maxi≤dxi≤ℓ(x)≤∑i=1dxi\mathop{\max }\limits_{i\le d}{x}_{i}\le \ell \left(x)\le \mathop{\sum }\limits_{i=1}^{d}{x}_{i}holds for any dd-variate STDF ℓ\ell , and therefore, 1≤ℓ(1d)≤d.1\le \ell \left({{\bf{1}}}_{d})\le d.Theorem 1Let ℓ\ell be a dd-variate STDF. Then, (i)ℓ(1d)=1⇔ℓ(x)=maxi≤dxi∀x\ell \left({{\bf{1}}}_{d})=1\iff \ell \left(x)={\max }_{i\le d}{x}_{i}\hspace{1.0em}\forall x(ii)ℓ(1d)=d⇔ℓ(x)=∑i=1dxi∀x\ell \left({{\bf{1}}}_{d})=d\iff \ell \left(x)={\sum }_{i=1}^{d}{x}_{i}\hspace{1.0em}\forall x.Proof(i)If ℓ(1d)=1\ell \left({{\bf{1}}}_{d})=1, then ℓ(ei)=∫widν(w)=1\ell \left({e}_{i})=\int {w}_{i}{\rm{d}}\nu \left(w)=1; hence, ν({w∈Cd∣wi=1})=1∀i≤d\nu \left(\left\{w\in {C}_{d}| {w}_{i}=1\right\})=1\hspace{1em}\forall i\le d, i.e., ν({1d})=1\nu \left(\left\{{{\bf{1}}}_{d}\right\})=1and ℓ(x)=\ell \left(x)=maxi≤dxi{\max }_{i\le d}{x}_{i}.(ii)If ℓ(1d)=d\ell \left({{\bf{1}}}_{d})=d, then ∫widν(w)=1d\int {w}_{i}{\rm{d}}\nu \left(w)=\frac{1}{d}for each ii; hence, ∫∑i=1dwidν(w)=1=∫maxi≤dwidν(w),\int \mathop{\sum }\limits_{i=1}^{d}{w}_{i}{\rm{d}}\nu \left(w)=1=\int \mathop{\max }\limits_{i\le d}{w}_{i}{\rm{d}}\nu \left(w),i.e., ∑i=1dwi=maxi≤dwi=1ν{\sum }_{i=1}^{d}{w}_{i}={\max }_{i\le d}{w}_{i}=1\hspace{0.33em}\nu -a.s., or ν({e1,…,ed})=1\nu \left(\left\{{e}_{1},\ldots ,{e}_{d}\right\})=1. From 1d=∫widν(w)=ν({ei})∀i≤d,\frac{1}{d}=\int {w}_{i}{\rm{d}}\nu \left(w)=\nu \left(\left\{{e}_{i}\right\})\hspace{1.0em}\forall i\le d,we deduce ν=1d∑i=1dεei\nu =\frac{1}{d}{\sum }_{i=1}^{d}{\varepsilon }_{{e}_{i}}, or ℓ(x)=∑i=1dxi\ell \left(x)={\sum }_{i=1}^{d}{x}_{i}.□Let X=(X1,…,Xd)X=\left({X}_{1},\ldots ,{X}_{d})have the MEV-distribution associated with the STDF ℓ\ell , i.e., with d.f. F(x)=exp−ℓ1xF\left(x)=\exp \left[-\ell \left(\frac{1}{x}\right)\right], for x∈]0,∞]dx\in {]0,\infty ]}^{d}. For ∅≠α⊆[d]\varnothing \ne \alpha \subseteq \left[d], the subvector Xα≔(Xi,i∈α){X}_{\alpha }:= \left({X}_{i},i\in \alpha )then has the d.f.Fα(xα)=F(xα,∞−α)=exp[−ℓ(xα−1,0−α)],{F}_{\alpha }\left({x}_{\alpha })=F\left({x}_{\alpha },{\infty }_{-\alpha })=\exp {[}-\ell ({x}_{\alpha }^{-1},{{\bf{0}}}_{-\alpha })],including F{i}(xi)=exp−1xi{F}_{\left\{i\right\}}\left({x}_{i})=\exp \left(-\frac{1}{{x}_{i}}\right), i=1,…,di=1,\ldots ,d.Condition (i) in Theorem 1 means that a.s. X1=X2=⋯=Xd{X}_{1}={X}_{2}=\cdots ={X}_{d}, and (ii) is equivalent with X1,…,Xd{X}_{1},\ldots ,{X}_{d}being iid (standard Fréchet). The independence of two subvectors of XXalso depends only on the extremal coefficients, as we now shall see.Theorem 2For disjoint (non-empty) subsets α,β⊆[d]\alpha ,\beta \subseteq \left[d], the following properties are equivalent: (i)ℓ(α)+ℓ(β)=ℓ(α∪β)\ell \left(\alpha )+\ell \left(\beta )=\ell \left(\alpha \cup \beta )(ii)ℓ(xα,0−α)+ℓ(xβ,0−β)=ℓ(xα∪β,0−(α∪β))∀x\ell ({x}_{\alpha },{{\bf{0}}}_{-\alpha })+\ell ({x}_{\beta },{{\bf{0}}}_{-\beta })=\ell ({x}_{\alpha \cup \beta },{{\bf{0}}}_{-\left(\alpha \cup \beta )})\hspace{1em}\forall x(iii)Xα{X}_{\alpha }and Xβ{X}_{\beta }are independent.ProofIn view of the connection between ℓ\ell and FF, only (i) ⇒\Rightarrow (ii) has to be shown. Without restriction α∪β=[d]\alpha \cup \beta =\left[d], i.e., β=−α\beta =-\alpha . So, let us assume (i), then from ℓ(α)=ℓ(1d)∫maxi∈αwidν(w)ℓ(β)=ℓ(1d)∫maxi∈βwidν(w),\begin{array}{rcl}\ell \left(\alpha )& =& \ell \left({{\bf{1}}}_{d})\displaystyle \int \mathop{\max }\limits_{i\in \alpha }{w}_{i}{\rm{d}}\nu \left(w)\\ \ell \left(\beta )& =& \ell \left({{\bf{1}}}_{d})\displaystyle \int \mathop{\max }\limits_{i\in \beta }{w}_{i}{\rm{d}}\nu \left(w),\end{array}we obtain ∫(maxi∈αwi+maxi∈βwi)dν(w)=∫maxi≤dwidν(w)=1.\int \left(\mathop{\max }\limits_{i\in \alpha }{w}_{i}+\mathop{\max }\limits_{i\in \beta }{w}_{i}\right){\rm{d}}\nu \left(w)=\int \mathop{\max }\limits_{i\le d}{w}_{i}{\rm{d}}\nu \left(w)=1.Let f(w)≔maxi∈αwif\left(w):= {\max }_{i\in \alpha }{w}_{i}, g(w)≔maxi∈βwig\left(w):= {\max }_{i\in \beta }{w}_{i}, w∈Cdw\in {C}_{d}. Then 0≤f≤10\le f\le 1, 0≤g≤10\le g\le 1, f∨g=1f\vee g=1, ∫(f+g)dν=∫f∨gdν\int (f+g){\rm{d}}\nu =\int f\vee g{\rm{d}}\nu . Since f∨g+f∧g=f+gf\vee g+f\wedge g=f+g, we obtain ∫f∧gdν=0\int f\wedge g{\rm{d}}\nu =0, or ν({f>0}∩{g>0})=0\nu (\{f\gt 0\}\cap \left\{g\gt 0\right\})=0, and Cd={f=1}∪{g=1}{C}_{d}=\{f=1\}\cup \left\{g=1\right\}. It follows ℓ(x)=ℓ(1d)∫maxi≤d(xiwi)dν(w)=ℓ(1d)⋅∫{f=1}…+∫{g=1}…=ℓ(xα,0−α)+ℓ(xβ,0−β).□\hspace{5em}\ell \left(x)=\ell \left({{\bf{1}}}_{d})\int \mathop{\max }\limits_{i\le d}\left({x}_{i}{w}_{i}){\rm{d}}\nu \left(w)=\ell \left({{\bf{1}}}_{d})\cdot \left(\mathop{\int }\limits_{\{f=1\}}\ldots +\mathop{\int }\limits_{\left\{g=1\right\}}\ldots \right)=\ell \left({x}_{\alpha },{{\bf{0}}}_{-\alpha })+\ell \left({x}_{\beta },{{\bf{0}}}_{-\beta }).\hspace{5em}\square Before we extend Theorem 2 to more than two subvectors, we need the following.Lemma 2Let I⊆R¯I\subseteq \overline{{\mathbb{R}}}be any non-degenerate interval, f:Id→Ra1d-↕f:{I}^{d}\to {\mathbb{R}}\hspace{0.33em}a\hspace{0.33em}{{\bf{1}}}_{d}\hspace{-0.33em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.3em}\updownarrow function, and [d]=α1∪α2∪⋯∪αn\left[d]={\alpha }_{1}\cup {\alpha }_{2}\cup \cdots \cup {\alpha }_{n}a partition with non-empty α1,…,αn{\alpha }_{1},\ldots ,{\alpha }_{n}. Define g:In→Rg:{I}^{n}\to {\mathbb{R}}by g(x1,…,xn)≔f∑j=1nxj1αjg\left({x}_{1},\ldots ,{x}_{n}):= f\left({\sum }_{j=1}^{n}{x}_{j}{{\bf{1}}}_{{\alpha }_{j}}\right). Then ggis 1n-↕{{\bf{1}}}_{n}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow . If ffis homogeneous, so is gg.Proofd=2d=2being trivial, assume d≥3d\ge 3. It is clearly enough to consider the partition α1={1},…,αd−2={d−2},αd−1={d−1,d},{\alpha }_{1}=\left\{1\right\},\ldots ,{\alpha }_{d-2}=\left\{d-2\right\},{\alpha }_{d-1}=\left\{d-1,d\right\},since the general case then follows easily by iteration.We have g(x1,…,xd−1)=f(x1,…,xd−2,xd−1,xd−1)g\left({x}_{1},\ldots ,{x}_{d-1})=f\left({x}_{1},\ldots ,{x}_{d-2},{x}_{d-1},{x}_{d-1}). Let x,z∈Id−1x,z\in {I}^{d-1}with x≤zx\le z, define x′≔(x1,…,xd−2)x^{\prime} := \left({x}_{1},\ldots ,{x}_{d-2}), and z′≔(z1,…,zd−2)z^{\prime} := \left({z}_{1},\ldots ,{z}_{d-2}), then by Lemma 1, Dzxg=Dz′x′g(⋯,xd−1)−Dz′x′g(⋯,zd−1)=Dz′x′f(⋯,xd−1,xd−1)−Dz′x′f(⋯,zd−1,zd−1)=[Dz′x′f(⋯,xd−1,xd−1)−Dz′x′f(⋯,zd−1,xd−1)]+[Dz′x′f(⋯,zd−1,xd−1)−Dz′x′f(⋯,zd−1,zd−1)]=Dzxf(⋯,⋅,xd−1)+Dzxf(⋯,zd−1,⋅)≤0\begin{array}{rcl}{D}_{z}^{x}g& =& {D}_{z^{\prime} }^{x^{\prime} }g\left(\cdots \hspace{0.33em},{x}_{d-1})-{D}_{z^{\prime} }^{x^{\prime} }g\left(\cdots \hspace{0.33em},{z}_{d-1})\\ & =& {D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{x}_{d-1},{x}_{d-1})-{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{z}_{d-1},{z}_{d-1})\\ & =& {[}{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{x}_{d-1},{x}_{d-1})-{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{z}_{d-1},{x}_{d-1})]+{[}{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{z}_{d-1},{x}_{d-1})-{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{z}_{d-1},{z}_{d-1})]\\ & =& {D}_{z}^{x}f\left(\cdots \hspace{0.33em},\cdot ,{x}_{d-1})+{D}_{z}^{x}f\left(\cdots \hspace{0.33em},{z}_{d-1},\cdot )\\ & \le & 0\end{array}as the sum of two non-positive numbers.□The following result is a considerable generalisation of Theorem 1 (ii) and Theorem 2.Theorem 3Let ℓ\ell be a dd-variate STDF, and let α1,…,αn{\alpha }_{1},\ldots ,{\alpha }_{n}be disjoint non-empty subsets of [d]\left[d]. The random vector X=(X1,…,Xd)X=\left({X}_{1},\ldots ,{X}_{d})is supposed to have the d.f. exp−ℓ1x\exp \left[-\ell \left(\frac{1}{x}\right)\right], x∈R+dx\in {{\mathbb{R}}}_{+}^{d}. Then the following conditions are equivalent: (i)ℓ(⋃j=1nαj)=∑j=1nℓ(αj)\ell ({\bigcup }_{j=1}^{n}{\alpha }_{j})={\sum }_{j=1}^{n}\ell \left({\alpha }_{j}).(ii){Xαj∣j=1,…,n}\left\{{X}_{{\alpha }_{j}}| j=1,\ldots ,n\right\}are independent.(iii){Xαj∣j=1,…,n}\left\{{X}_{{\alpha }_{j}}| j=1,\ldots ,n\right\}are pairwise independent.ProofWithout restriction, we assume ⋃j=1nαj=[d]{\bigcup }_{j=1}^{n}{\alpha }_{j}=\left[d].(i) ⇒\Rightarrow (ii): We use induction, the case n=2n=2being true by Theorem 2. Supposing the conclusion for nn, we use ℓ\ell ’s subadditivity to obtain ℓ⋃j=1n+1αj≤ℓ⋃j=1nαj+ℓ(αn+1)≤∑j=1nℓ(αj)+ℓ(αn+1)=ℓ⋃j=1n+1αj,\ell \left(\mathop{\bigcup }\limits_{j=1}^{n+1}{\alpha }_{j}\right)\le \ell \left(\mathop{\bigcup }\limits_{j=1}^{n}{\alpha }_{j}\right)+\ell \left({\alpha }_{n+1})\le \mathop{\sum }\limits_{j=1}^{n}\ell \left({\alpha }_{j})+\ell \left({\alpha }_{n+1})=\ell \left(\mathop{\bigcup }\limits_{j=1}^{n+1}{\alpha }_{j}\right),and hence, ℓ(⋃j=1nαj)=∑j=1nℓ(αj)\ell ({\bigcup }_{j=1}^{n}{\alpha }_{j})={\sum }_{j=1}^{n}\ell \left({\alpha }_{j}), {Xαj∣j≤n}\left\{{X}_{{\alpha }_{j}}| j\le n\right\}are independent, and n=2n=2may be applied to ⋃j=1nαj{\bigcup }_{j=1}^{n}{\alpha }_{j}and αn+1{\alpha }_{n+1}.(iii) ⇒\Rightarrow (i): We use again induction. For n=2n=2, there is nothing to prove. We assume validity for some n≥2n\ge 2and consider the case n+1n+1. Let α≔⋃j<nαj,β≔αn,γ≔αn+1\alpha := \bigcup _{j\lt n}{\alpha }_{j},\hspace{1.0em}\beta := {\alpha }_{n},\hspace{1.0em}\gamma := {\alpha }_{n+1}and define ffon R+3{{\mathbb{R}}}_{+}^{3}by f(a,b,c)≔ℓ(a1α+b1β+c1γ)f\left(a,b,c):= \ell \left(a{{\bf{1}}}_{\alpha }+b{{\bf{1}}}_{\beta }+c{{\bf{1}}}_{\gamma }). By Lemma 2, ffis 13-↕{{\bf{1}}}_{3}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow (not normalised!). Therefore, 0≥D1303f=f(0)−∑f(ei)+∑i<jf(ei+ej)−f(13)=0−ℓ(α)−ℓ(β)−ℓ(γ)+ℓ(α∪β)+ℓ(α∪γ)+ℓ(β∪γ)−ℓ(α∪β∪γ)=−∑j<nℓ(αj)−ℓ(αn)−ℓ(αn+1)+∑j≤nℓ(αj)+∑j≠nℓ(αj)+(ℓ(αn)+ℓ(αn+1))−ℓ⋃j=1n+1αj,\begin{array}{rcl}0& \ge & {D}_{{{\bf{1}}}_{3}}^{{{\bf{0}}}_{3}}f=f\left(0)-\displaystyle \sum f\left({e}_{i})+\displaystyle \sum _{i\lt j}f\left({e}_{i}+{e}_{j})-f\left({{\bf{1}}}_{3})\\ & =& 0-\ell \left(\alpha )-\ell \left(\beta )-\ell \left(\gamma )+\ell \left(\alpha \cup \beta )+\ell \left(\alpha \cup \gamma )+\ell \left(\beta \cup \gamma )-\ell \left(\alpha \cup \beta \cup \gamma )\\ & =& -\displaystyle \sum _{j\lt n}\ell \left({\alpha }_{j})-\ell \left({\alpha }_{n})-\ell \left({\alpha }_{n+1})+\displaystyle \sum _{j\le n}\ell \left({\alpha }_{j})+\displaystyle \sum _{j\ne n}\ell \left({\alpha }_{j})+\left(\ell \left({\alpha }_{n})+\ell \left({\alpha }_{n+1}))-\ell \left(\mathop{\bigcup }\limits_{j=1}^{n+1}{\alpha }_{j}\right),\end{array}i.e., ∑j=1n+1ℓ(αj)≤ℓ⋃j=1n+1αj≤∑j=1n+1ℓ(αj).□\hspace{16em}\mathop{\sum }\limits_{j=1}^{n+1}\ell \left({\alpha }_{j})\le \ell \left(\mathop{\bigcup }\limits_{j=1}^{n+1}{\alpha }_{j}\right)\le \mathop{\sum }\limits_{j=1}^{n+1}\ell \left({\alpha }_{j}).\hspace{16.5em}\square Considering in Theorem 3 the special case αj={j}{\alpha }_{j}=\left\{j\right\}, j=1,…,dj=1,\ldots ,d, we are back to Theorem 1 (ii), with the additional equivalence to pairwise independence, i.e., ℓ(ei+ej)=2∀i≠j\ell \left({e}_{i}+{e}_{j})=2\hspace{1em}\forall i\ne j. One might be tempted to believe that there is a corresponding generalisation of part (i) of Theorem 1 as well. This is not the case.Theorem 4(i)Let ℓ\ell be a d-variate STDF; α,β⊆[d]\alpha ,\beta \subseteq \left[d]such that α∩β≠∅\alpha \cap \beta \ne \varnothing . If ℓ(α)=ℓ(β)=1\ell \left(\alpha )=\ell \left(\beta )=1, then also ℓ(α∪β)=1\ell \left(\alpha \cup \beta )=1.(ii)Let f,gf,gbe mm- (resp. nn-)variate STDFs, such that also ℓ(x,y)≔f(x)∨g(y)\ell \left(x,y):= f\left(x)\vee g(y)is a STDF. Then f=maxf=\max , g=maxg=\max (and ℓ=max\ell =\max ).Proof(i)Let XXbe a R+d{{\mathbb{R}}}_{+}^{d}-valued random vector with STDF ℓ\ell . Then, if ℓ(α)=ℓ(β)=1,Xi=Xj\ell \left(\alpha )=\ell \left(\beta )=1,{X}_{i}={X}_{j}a.s. ∀i,j∈α\forall \hspace{-0.3em}i,j\in \alpha and ∀i,j∈β\forall \hspace{-0.3em}i,j\in \beta , and because of α∩β≠∅\alpha \cap \beta \ne \varnothing , Xi=Xj{X}_{i}={X}_{j}a.s. ∀i,j∈α∪β\forall \hspace{-0.3em}i,j\in \alpha \cup \beta . That is, ℓ(α∪β)=1\ell \left(\alpha \cup \beta )=1.(ii)Again let (X1,…,Xm,Y1,…,Yn)\left({X}_{1},\ldots ,{X}_{m},{Y}_{1},\ldots ,{Y}_{n})have ℓ\ell as its STDF. Then, Xi=Yj{X}_{i}={Y}_{j}a.s. ∀i∈[m]\forall \hspace{-0.3em}i\in \left[m], ∀j∈[n]\forall \hspace{-0.3em}j\in \left[n], i.e., X1=⋯=Xm=Y1=⋯=Yn{X}_{1}=\cdots ={X}_{m}={Y}_{1}=\cdots ={Y}_{n}a.s., leading to ℓ=max\ell =\max , f=maxf=\max and g=maxg=\max .□Remark 2For “overlapping variables”, this is different: ℓ(x,y,z)≔(x+y)∨(y+z)=x∨z+y\ell \left(x,y,z):= \left(x+y)\vee (y+z)=x\vee z+yis a STDF, as is also (with a,b∈[0,1]a,b\in \left[0,1]) ℓ(x,y)≔(ax+y)∨(x+by)=ax+by+[(1−a)x]∨[(1−b)y].\ell \left(x,y):= \left(ax+y)\vee \left(x+by)=ax+by+\left[\left(1-a)x]\vee \left[\left(1-b)y].With iid standard Fréchet random variables XX, YY, and ZZ, a stochastic model for these two STDFs would be the random vector (X,Y,X)\left(X,Y,X), resp. ((aX)∨(1−a)Z,(bY)∨(1−b)Z)\left(\left(aX)\vee \left(1-a)Z,\left(bY)\vee \left(1-b)Z).Note, however, that f(x,y,z)≔(x+y)∨(y+z)∨(z+x)f\left(x,y,z):= \left(x+y)\vee (y+z)\vee \left(z+x)is not a STDF: D1303f=0−3+6−2=1>0{D}_{{{\bf{1}}}_{3}}^{{{\bf{0}}}_{3}}f=0-3+6-2=1\gt 0.4Characterisation of logistic and related STDFsPerhaps the best-known STDFs are the logistic ones, i.e., the family {ℓp∣p∈[1,∞[}\left\{{\ell }_{p}| p\in {[}1,\infty {[}\right\}, defined by ℓp(x)≔∑i=1dxip1/p,x∈R+d.{\ell }_{p}\left(x):= {\left(\mathop{\sum }\limits_{i=1}^{d}{x}_{i}^{p}\right)}^{1\text{/}p},\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d}.Among all symmetric STDFs they are particular, depending on xxin an “additive way,” being a function of ∑i=1dg(xi){\sum }_{i=1}^{d}g\left({x}_{i})for some g:R+→R+g:{{\mathbb{R}}}_{+}\to {{\mathbb{R}}}_{+}. We shall see that there are no other STDFs with this property besides the logistic ones.We begin by solving a functional equation.Theorem 5Let φ:R+2→R+\varphi :{{\mathbb{R}}}_{+}^{2}\to {{\mathbb{R}}}_{+}be homogeneous, ≢0\not\equiv 0, and let g:[0,∞]→[0,∞]g:\left[0,\infty ]\to \left[0,\infty ]be a continuous bijection, such that g(1)=1g\left(1)=1andg(φ(x,y))=g(x)+g(y)∀x,y∈R+.g\left(\varphi \left(x,y))=g\left(x)+g(y)\hspace{1.0em}\forall x,y\in {{\mathbb{R}}}_{+}.Then, ∃p∈R⧹{0}\exists \hspace{-0.16em}p\in {\mathbb{R}}\setminus \left\{0\right\}such that g(x)=xp∀x∈]0,∞[g\left(x)={x}^{p}\hspace{1em}\forall x\in ]0,\infty {[}(which of course extends uniquely to [0,∞]\left[0,\infty ]).ProofObviously g(]0,∞[)=]0,∞[g\left(]0,\infty {[})=]0,\infty {[}and g({0,∞})={0,∞}g\left(\left\{0,\infty \right\})=\left\{0,\infty \right\}, and ggis either (strictly) increasing or decreasing. Since g−1{g}^{-1}is also continuous, so is φ\varphi .For a≔φ(1,1)a:= \varphi \left(1,1), we have g(a)=2g\left(a)=2and g(ta)=g(t⋅φ(1,1))=g(φ(t,t))=2g(t)∀t∈R+g\left(ta)=g\left(t\cdot \varphi \left(1,1))=g\left(\varphi \left(t,t))=2g\left(t)\hspace{1em}\forall t\in {{\mathbb{R}}}_{+}(i.e., g(0)=2g(0)g\left(0)=2g\left(0), in accordance with g(0)∈{0,∞}g\left(0)\in \left\{0,\infty \right\}). The equality g(ta)=g(t)g(a)g\left(ta)=g\left(t)g\left(a)shows aato belong to G≔{x∈]0,∞[∣g(tx)=g(t)g(x)∀t∈]0,∞[},G:= \left\{x\in ]0,\infty {[}| g\left(tx)=g\left(t)g\left(x)\hspace{1em}\forall t\in ]0,\infty {[}\right\},a multiplicative subgroup of ]0,∞[]0,\infty {[}as is easily seen. Hence, {an∣n∈Z}⊆G\left\{{a}^{n}| n\in {\mathbb{Z}}\right\}\subseteq G.For n∈Zn\in {\mathbb{Z}}, g(φ(1,an))=1+g(an)=1+(g(a))n=1+2n,g\left(\varphi \left(1,{a}^{n}))=1+g\left({a}^{n})=1+{\left(g\left(a))}^{n}=1+{2}^{n},and for t>0t\gt 0, g(t⋅φ(1,an))=g(φ(t,tan))=g(t)+g(tan)=g(t)(1+2n)=g(t)g(φ(1,an)),g\left(t\cdot \varphi \left(1,{a}^{n}))=g\left(\varphi \left(t,t{a}^{n}))=g\left(t)+g\left(t{a}^{n})=g\left(t)\left(1+{2}^{n})=g\left(t)g\left(\varphi \left(1,{a}^{n})),i.e., also {φ(1,an)∣n∈Z}⊆G\left\{\varphi \left(1,{a}^{n})| n\in {\mathbb{Z}}\right\}\subseteq G, where φ(1,an)=g−1(1+2n)\varphi \left(1,{a}^{n})={g}^{-1}\left(1+{2}^{n}), and this converges to g−1(1)=1{g}^{-1}\left(1)=1for n→−∞n\to -\infty .This implies GGto be dense in ]0,∞[]0,\infty {[}: it suffices to show 1<u<v⇒G∩]u,v[≠∅,1\lt u\lt v\Rightarrow G\cap ]u,v{[}\hspace{0.33em}\ne \hspace{0.33em}\varnothing ,and this follows because for any x∈1,vux\in \left]1,\frac{v}{u}\right[, {xj∣j∈N}∩]u,v[≠∅\left\{{x}^{j}| j\in {\mathbb{N}}\right\}\cap ]u,v{[}\hspace{0.33em}\ne \hspace{0.33em}\varnothing (choose kkwith xk−1≤u<xk{x}^{k-1}\le u\lt {x}^{k}, then xk=x⋅xk−1<vu⋅u=v{x}^{k}=x\cdot {x}^{k-1}\lt \frac{v}{u}\cdot u=v). If ggis increasing, then g−1(1+2n)∈1,vu{g}^{-1}\left(1+{2}^{n})\in \left]1,\frac{v}{u}\right[for some (negative!) nn, and for decreasing gg, we may choose instead [g−1(1+2n)]−1{\left[{g}^{-1}\left(1+{2}^{n})]}^{-1}.Now GGis closed, ggbeing continuous; hence, G=]0,∞[G=]0,\infty {[}and g(xy)=g(x)g(y)∀x,y∈]0,∞[g\left(xy)=g\left(x)g(y)\hspace{1em}\forall x,y\in ]0,\infty {[}. It is well known that this implies g(x)=xpg\left(x)={x}^{p}for some p≠0p\ne 0. (For f≔log∘g∘exp:R→Rf:= \log \circ g\circ \exp :{\mathbb{R}}\to {\mathbb{R}}, we have f(s+t)=f(s)+f(t)∀s,tf\left(s+t)=f\left(s)+f\left(t)\hspace{1em}\forall s,t; this is the standard Cauchy equation, and ffbeing continuous, it has the form f(s)=c⋅sf\left(s)=c\cdot swith c∈Rc\in {\mathbb{R}}; therefore, g(x)=xcg\left(x)={x}^{c}.) From g(a)=2=apg\left(a)=2={a}^{p}, we obtain p=log2loga>0fora>1<0fora<1.p=\frac{\log 2}{\log a}\left\{\begin{array}{l}\gt 0\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}a\gt 1\hspace{1.0em}\\ \lt 0\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}a\lt 1.\hspace{1.0em}\end{array}\right.□Theorem 6(Characterisation of logistic STDFs) Let ℓ\ell be a d-variate STDF of the formℓ(x)=g−1∑i=1dg(xi)\ell \left(x)={g}^{-1}\left(\mathop{\sum }\limits_{i=1}^{d}g\left({x}_{i})\right)for some continuous bijection g:R+→R+g:{{\mathbb{R}}}_{+}\to {{\mathbb{R}}}_{+}, g(1)=1g\left(1)=1without restriction.Then, g(x)=xpg\left(x)={x}^{p}for some p≥1p\ge 1, i.e., ℓ=ℓp\ell ={\ell }_{p}.ProofObviously ggis (strictly) increasing, in particular g(0)=0g\left(0)=0, and it suffices to consider d=2d=2. By the preceding theorem, g(x)=xpg\left(x)={x}^{p}for some p∈]0,∞[p\in ]0,\infty {[}, and g(ℓ(1,1))=(ℓ(1,1))p=2g\left(\ell \left(1,1))={\left(\ell \left(1,1))}^{p}=2implies ℓ(1,1)>1\ell \left(1,1)\gt 1, and p=log2logℓ(1,1)≥1p=\frac{\log 2}{\log \ell \left(1,1)}\ge 1since ℓ(1,1)≤2\ell \left(1,1)\le 2.□This result, assuming from the outset (though tacitly) the function ggto be differentiable, was shown in an equivalent form for copulas, stating that the only Archimedean extreme value copulas are the logistic (or Gumbel) ones, cf. [3]. We state this as a corollary, being slightly more general while not assuming differentiability:Corollary 1Let CCbe a dd-variate Archimedean copula, i.e., C(u)=h−1∑i=1dh(ui),u∈]0,1]dC\left(u)={h}^{-1}\left(\mathop{\sum }\limits_{i=1}^{d}h\left({u}_{i})\right),\hspace{1.0em}u\in {]0,1]}^{d}with a decreasing bijection h:]0,1]→R+h:]0,1]\to {{\mathbb{R}}}_{+}, and assume that CCis also “extreme”, i.e., C(u1t,…,udt)=(C(u))t∀t>0,∀u∈]0,1]d.C\left({u}_{1}^{t},\ldots ,{u}_{d}^{t})={\left(C\left(u))}^{t}\hspace{1em}\forall t\gt 0,\hspace{1em}\forall u\in {]0,1]}^{d}.Then, h(s)=(−logs)ph\left(s)={\left(-\log s)}^{p}for some p≥1p\ge 1.ProofWe only need to consider d=2d=2. It is easy to see that C(u)>0C\left(u)\gt 0for u∈]0,1]2u\in {]0,1]}^{2}. The corresponding STDF ℓ\ell is given as follows: ℓ(x,y)=−logC(e−x,e−y)=g−1(g(x)+g(y)),\ell \left(x,y)=-\log C\left({e}^{-x},{e}^{-y})={g}^{-1}\left(g\left(x)+g(y)),where g(x)≔h(e−x)g\left(x):= h\left({e}^{-x}). The preceding theorem implies g(x)=xpg\left(x)={x}^{p}for some p≥1p\ge 1; hence, h(s)=(−logs)ph\left(s)={\left(-\log s)}^{p}.□Remark 3In the definition of an Archimedean copula (as in Nelsen’s book [4]), it is not assumed that hhis unbounded; the case of a decreasing bijection h:]0,1]→[0,a[h:]0,1]\to {[}0,a{[}for some finite aais also allowed, with h−1{h}^{-1}extended to R+{{\mathbb{R}}}_{+}by h−1(x)≔0∀x≥a{h}^{-1}\left(x):= 0\hspace{1em}\forall x\ge a. But then a copula of the form C(u,v)=h−1(h(u)+h(v))C\left(u,v)={h}^{-1}\left(h\left(u)+h\left(v))cannot be extreme: choose u,v>0u,v\gt 0such that h(u)>a2h\left(u)\gt \frac{a}{2}, h(v)>a2h\left(v)\gt \frac{a}{2}, so that h(u)+h(v)>ah\left(u)+h\left(v)\gt a. Then ut→1{u}^{t}\to 1, vt→1{v}^{t}\to 1for t→0t\to 0, h(ut)→0h\left({u}^{t})\to 0, h(vt)→0h\left({v}^{t})\to 0and h−1(h(ut)+h(vt))→h−1(0)=1,t→0.{h}^{-1}(h\left({u}^{t})+h\left({v}^{t}))\to {h}^{-1}\left(0)=1,t\to 0.However, (h−1(h(u)+h(v)))t=0∀t>0.{({h}^{-1}\left(h\left(u)+h\left(v)))}^{t}=0\hspace{1.0em}\forall t\gt 0.Also so-called negative logistic (or Galambos) STDFs can be characterised by an “Archimedean property”, which however is not obvious at first sight. We remind that any STDF ℓ\ell on R+d{{\mathbb{R}}}_{+}^{d}is the co-survival function of some homogeneous Radon measure μ\mu on the locally compact space [0,∞]d⧹{∞d}≕Zd{\left[0,\infty ]}^{d}\setminus \left\{{\infty }_{d}\right\}\hspace{0.33em}=: \hspace{0.33em}{Z}_{d}, i.e., ℓ(x)=μ([x,∞]c)≕μˇ(x),x∈R+d.\ell \left(x)=\mu ({\left[x,\infty ]}^{c})\hspace{0.33em}=: \hspace{0.33em}\check{\mu }\left(x),\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d}.By definition of a Radon measure, μˇ(x)<∞∀x\check{\mu }\left(x)\lt \infty \hspace{1em}\forall x. The d.f. μ^(x)≔μ([0,x])\widehat{\mu }\left(x):= \mu \left(\left[0,x])of μ\mu is of course also finite and homogeneous.The family {fp∣p∈]−∞,0[}\{{f}_{p}| p\in ]-\infty ,0{[}\}of negative logistic STDFs is defined by fp(x)≔∑∅≠α⊆[d](−1)∣α∣+1∑i∈αxip1/p,x∈R+d.{f}_{p}\left(x):= \sum _{\varnothing \ne \alpha \subseteq \left[d]}{\left(-1)}^{| \alpha | +1}{\left(\sum _{i\in \alpha }{x}_{i}^{p}\right)}^{1\text{/}p},\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d}.Theorem 7Let ℓ\ell be a d-variate STDF, ℓ=μˇ\ell =\check{\mu }, with μ∈M+(Zd)\mu \in {M}_{+}\left({Z}_{d}), such that the d.f. μ^\widehat{\mu }is “Archimedean,” i.e., μ^(x)=g−1∑i=1dg(xi),x∈R+d,\widehat{\mu }\left(x)={g}^{-1}\left(\mathop{\sum }\limits_{i=1}^{d}g\left({x}_{i})\right),\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d},where g:[0,∞]→[0,∞]g:\left[0,\infty ]\to \left[0,\infty ]is a continuous bijection. Then, ∃p<0\exists \hspace{-0.16em}p\lt 0such that g(x)=xpg\left(x)={x}^{p}, and ℓ=fp\ell ={f}_{p}.ProofBy assumption μ^(x1,…,xd−1,∞)=limxd→∞g−1∑i<dg(xi)+g(xd)\widehat{\mu }\left({x}_{1},\ldots ,{x}_{d-1},\infty )=\mathop{\mathrm{lim}}\limits_{{x}_{d}\to \infty }{g}^{-1}\left(\sum _{i\lt d}g\left({x}_{i})+g\left({x}_{d})\right)is finite; hence, g(∞)=0g\left(\infty )=0and ggis decreasing. By iteration, we obtain μ^(x1,x2,∞,…,∞)=g−1(g(x1)+g(x2)),\widehat{\mu }\left({x}_{1},{x}_{2},\infty ,\ldots ,\infty )={g}^{-1}\left(g\left({x}_{1})+g\left({x}_{2})),and from Theorem 5, we infer g(x)=xpg\left(x)={x}^{p}, where p<0p\lt 0, i.e., μ^(x)=∑i≤dxip1/p∀x∈R+d\widehat{\mu }\left(x)={\left({\sum }_{i\le d}{x}_{i}^{p}\right)}^{1\text{/}p}\hspace{1em}\forall x\in {{\mathbb{R}}}_{+}^{d}.The co-survival function μˇ\check{\mu }of μ\mu is easily expressed in terms of μ^\widehat{\mu }(where we use that boundaries of intervals are μ\mu -null sets, μ\mu being homogeneous, see [6], p. 248): with Bi≔{y∈Zd∣yi<xi}{B}_{i}:= \{y\in {Z}_{d}| {y}_{i}\lt {x}_{i}\}, we have μˇ(x)=μ([x,∞]c)=μ(⋃i≤dBi)=∑iμ(Bi)−∑i<jμ(Bi∩Bj)±⋯,\check{\mu }\left(x)=\mu \left({\left[x,\infty ]}^{c})=\mu \left(\bigcup _{i\le d}{B}_{i}\right)=\sum _{i}\mu \left({B}_{i})-\sum _{i\lt j}\mu \left({B}_{i}\cap {B}_{j})\pm \cdots \hspace{0.33em},and since μ(B1)=μ^(x1,∞,∞,…)\mu \left({B}_{1})=\widehat{\mu }\left({x}_{1},\infty ,\infty ,\ldots )etc., μ(B1∩B2)=μ^(x1,x2,∞,∞,…)\mu \left({B}_{1}\cap {B}_{2})=\widehat{\mu }\left({x}_{1},{x}_{2},\infty ,\infty ,\ldots )etc., we arrive at ℓ(x)=μˇ(x)=∑xi−∑i<j(xip+xjp)1/p±⋯=fp(x).□\hspace{10.2em}\ell \left(x)=\check{\mu }\left(x)=\sum {x}_{i}-\sum _{i\lt j}{({x}_{i}^{p}+{x}_{j}^{p})}^{1\text{/}p}\pm \cdots ={f}_{p}\left(x).\hspace{15em}\square Remark 4In the above theorem, we have μ^(x)=∑i=1dxip1/p≕μ^p(x).\widehat{\mu }\left(x)={\left(\mathop{\sum }\limits_{i=1}^{d}{x}_{i}^{p}\right)}^{1\text{/}p}\hspace{0.33em}=: \hspace{0.33em}{\widehat{\mu }}_{p}\left(x).In ref. [5], Theorem 6, it was shown that this function is a “bona fide” d.f. iff p∈−∞,1d−1∪1d−2,…,12,1p\in \left[-\infty ,\frac{1}{d-1}\right]\cup \left\{\frac{1}{d-2},\ldots ,\frac{1}{2},1\right\}. The question arises if for positive ppin this set, there is also a corresponding STDF: the answer is NO: μp{\mu }_{p}is then a Radon measure on R+d{{\mathbb{R}}}_{+}^{d}, not on Zd=[0,∞]d⧹{∞d}{Z}_{d}={\left[0,\infty ]}^{d}\setminus \left\{{\infty }_{d}\right\}, in fact μˇp(x)=∞{\check{\mu }}_{p}\left(x)=\infty for each x≠0x\ne 0in R+d{{\mathbb{R}}}_{+}^{d}. For p→0p\to 0, we obtain as limit μ^0(x)=(∏i=1dxi)1/d{\widehat{\mu }}_{0}\left(x)={({\prod }_{i=1}^{d}{x}_{i})}^{1\text{/}d}, and μ0{\mu }_{0}is likewise not a Radon measure on Zd{Z}_{d}.Remark 5Theorems 6 and 7 add to the many common features between Gumbel (logistic) and Galambos (negative logistic) STDFs resp. copulas, nicely described in ref. [2].We also want to characterise nested logistic STDFs, but here we are first confronted with the interesting general question of the “composebility” of several STDFs in its simplest (already non-trivial) form: if f,g,hf,g,hare bivariate STDFs, when is f∘(g×h)f\circ \left(g\times h)again a STDF? For logistic STDFs f=ℓr,g=ℓpf={\ell }_{r},g={\ell }_{p}and h=ℓqh={\ell }_{q}(r,p,q∈[1,∞[r,p,q\in {[}1,\infty {[}), a sufficient condition is well known: r≤pr\le pand r≤qr\le q. We shall show that this is necessary, too.Theorem 8Let r,p>0r,p\gt 0such thatℓ(x,y,z)≔xp+yppr+zrr\ell \left(x,y,z):= \sqrt[r]{{\sqrt[p]{{x}^{p}+{y}^{p}}}^{r}+{z}^{r}}is a STDF. Then, 1≤r≤p1\le r\le p.Proofr≥1r\ge 1and p≥1p\ge 1is clear. We start with 0≥D(1,1)(0,0)ℓ(⋅,⋅,t)=t−2(1+tr)1/r+(2r/p+tr)1/r0\ge {D}_{\left(1,1)}^{\left(0,0)}\ell \left(\cdot ,\cdot ,t)=t-2{\left(1+{t}^{r})}^{1\text{/}r}+{\left({2}^{r\text{/}p}+{t}^{r})}^{1\text{/}r}or (2r/p+tr)1/r+t≤2(1+tr)1/r{\left({2}^{r\text{/}p}+{t}^{r})}^{1\text{/}r}+t\le 2{\left(1+{t}^{r})}^{1\text{/}r}for any t≥0t\ge 0.Equivalently, 2r/p≤[2(1+tr)1/r−t]r−tr∀t>0.{2}^{r\text{/}p}\le {{[}2{\left(1+{t}^{r})}^{1\text{/}r}-t]}^{r}-{t}^{r}\hspace{1.0em}\forall t\gt 0.We now make use of the Binomial series (1+x)r=1+rx+r2x2+⋯{\left(1+x)}^{r}=1+rx+\left(\genfrac{}{}{0.0pt}{}{r}{2}\right){x}^{2}+\cdots valid for ∣x∣<1| x| \lt 1and all r∈Rr\in {\mathbb{R}}. We obtain [2(1+tr)1/r−t]r−tr=tr[2(1+t−r)1/r−1]r−tr=tr2(1+1r⋅t−r+1∕r2t−2r+⋯)−1r−tr=tr1+2r⋅t−r+21∕r2t−2r+⋯r−tr=tr1+2r⋅t−r(1+o(1))r−tr=tr1+2t−r(1+o(1))+r22(1+o(1))rtr2+⋯−tr=2(1+o(1))+O(1)/tr→2fort→∞.\begin{array}{rcl}{{[}2{\left(1+{t}^{r})}^{1\text{/}r}-t]}^{r}-{t}^{r}& =& {t}^{r}{{[}2{\left(1+{t}^{-r})}^{1\text{/}r}-1]}^{r}-{t}^{r}\\ & =& {t}^{r}{\left[2\left(1+\frac{1}{r}\cdot {t}^{-r}+\left(\genfrac{}{}{0.0pt}{}{1/r}{2}\right){t}^{-2r}+\cdots )-1\right]}^{r}-{t}^{r}\\ & =& {t}^{r}{\left[1+\frac{2}{r}\cdot {t}^{-r}+2\left(\genfrac{}{}{0.0pt}{}{1/r}{2}\right){t}^{-2r}+\cdots \right]}^{r}-{t}^{r}\\ & =& {t}^{r}{\left[1+\frac{2}{r}\cdot {t}^{-r}\left(1+{\mathcal{o}}\left(1))\right]}^{r}-{t}^{r}\\ & =& {t}^{r}\left[1+2{t}^{-r}\left(1+{\mathcal{o}}\left(1))+\left(\genfrac{}{}{0.0pt}{}{r}{2}\right){\left(\frac{2\left(1+{\mathcal{o}}\left(1))}{r{t}^{r}}\right)}^{2}+\cdots \hspace{0.33em}\right]-{t}^{r}\\ & =& 2\left(1+{\mathcal{o}}\left(1))+{\mathcal{O}}\left(1)\hspace{0.1em}\text{/}\hspace{0.1em}{t}^{r}\to 2\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}t\to \infty .\end{array}As a consequence, r/p≤1r\hspace{0.1em}\text{/}\hspace{0.1em}p\le 1, or r≤pr\le p.□We arrive at a complete characterisation for composite logistic STDFs.Theorem 9Let r,p1,…,pd∈[1,∞[r,{p}_{1},\ldots ,{p}_{d}\in {[}1,\infty {[}and α1∪⋯∪αd=[n]{\alpha }_{1}\cup \cdots \cup {\alpha }_{d}=\left[n]a partition, ∣αj∣≥2∀j| {\alpha }_{j}| \ge 2\hspace{1em}\forall j. Then ℓr∘(ℓp1×⋯×ℓpd){\ell }_{r}\circ \left({\ell }_{{p}_{1}}\times \cdots \times {\ell }_{{p}_{d}})is a STDF if and only if r≤pir\le {p}_{i}for all ii.ProofSufficiency is well known, see, e.g., ref. [8], p. 256. The other direction follows from the previous theorem by considering (for p1{p}_{1}) ℓr(ℓp1(xei+yej),ℓp2(z⋅ek),ℓp3(0),…),{\ell }_{r}\left({\ell }_{{p}_{1}}\left(x{e}_{i}+y{e}_{j}),{\ell }_{{p}_{2}}\left(z\cdot {e}_{k}),{\ell }_{{p}_{3}}\left(0),\ldots ),where i,j∈α1i,j\in {\alpha }_{1}and k∈α2k\in {\alpha }_{2}, which gives r≤p1r\le {p}_{1}.□The “nested” STDFs just considered allow the following “Archimedean” characterisation:Theorem 10Let f,g,hf,g,hbe continuous bijections of R+{{\mathbb{R}}}_{+}, m≥2m\ge 2, n≥2n\ge 2. Ifℓ(x,y)=f−1fg−1∑i≤mg(xi)+fh−1∑j≤nh(yj)\ell \left(x,y)={f}^{-1}\left\{f\left[{g}^{-1}\left(\sum _{i\le m}g\left({x}_{i})\right)\right]+f\left[{h}^{-1}\left(\sum _{j\le n}h({y}_{j})\right)\right]\right\}is a STDF on R+m×R+n{{\mathbb{R}}}_{+}^{m}\times {{\mathbb{R}}}_{+}^{n}, then ℓ=ℓr∘(ℓp×ℓq)\ell ={\ell }_{r}\circ \left({\ell }_{p}\times {\ell }_{q})for some r,p,q≥1r,p,q\ge 1, such that r≤pr\le pand r≤qr\le q(ℓr{\ell }_{r}bivariate).ProofPutting y=0y=0, we have ℓ(x,0)=g−1∑i≤mg(xi)\ell \left(x,0)={g}^{-1}\left({\sum }_{i\le m}g\left({x}_{i})\right); hence, g(s)=spg\left(s)={s}^{p}, where p≥1p\ge 1. Similarly, h(s)=sqh\left(s)={s}^{q}, with q≥1q\ge 1. For x=(x1,0,…,0)x=\left({x}_{1},0,\ldots ,0), y=(y1,0,…,0)y=({y}_{1},0,\ldots ,0), we obtain ℓ(x1,0,…,0,y1,0,…,0)=f−1(f(x1)+f(y1))\ell \left({x}_{1},0,\ldots ,0,{y}_{1},0,\ldots ,0)={f}^{-1}(f\left({x}_{1})+f({y}_{1})), and so f(s)=srf\left(s)={s}^{r}, where r≥1r\ge 1. By Theorem 9, finally, r≤pr\le pand r≤qr\le q.□Corollary 2For m,n≥2m,n\ge 2and decreasing bijections f,g,h:]0,1]→R+f,g,h:]0,1]\to {{\mathbb{R}}}_{+}, ifC(u,v)=f−1fg−1∑i≤mg(ui)+fh−1∑j≤nh(vj)C\left(u,v)={f}^{-1}\left\{f\left[{g}^{-1}\left(\sum _{i\le m}g\left({u}_{i})\right)\right]+f\left[{h}^{-1}\left(\sum _{j\le n}h\left({v}_{j})\right)\right]\right\}is an extreme value copula, thenC(u,v)=∑i(−logui)pr/p+∑j(−logvj)qr/q1/rC\left(u,v)={\left\{{\left[\sum _{i}{\left(-\log {u}_{i})}^{p}\right]}^{r\text{/}p}+{\left[\sum _{j}{\left(-\log {v}_{j})}^{q}\right]}^{r\text{/}q}\right\}}^{1\text{/}r}for some r,p,q≥1r,p,q\ge 1with r≤pr\le pand r≤qr\le q.This class of copulas of “composite Gumbel type” was already considered in ref. [7], p. 366, as particular examples of so-called generalised Archimedean copulas. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Dependence Modeling de Gruyter

Stable tail dependence functions – some basic properties

Dependence Modeling , Volume 10 (1): 11 – Jan 1, 2022

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de Gruyter
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© 2022 Paul Ressel, published by De Gruyter
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2300-2298
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2300-2298
DOI
10.1515/demo-2022-0114
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Abstract

1IntroductionA multivariate extreme value (MEV) distribution (in a standardised form) is given by a distribution function (“d.f.”) FFon R+d{{\mathbb{R}}}_{+}^{d}with the decisive property (F(tx))t=F(x)∀x∈R+d,∀t>0,{\left(F\left(tx))}^{t}=F\left(x)\hspace{1.0em}\forall x\in {{\mathbb{R}}}_{+}^{d},\hspace{1em}\forall t\gt 0,and with standard one-dimensional Fréchet margins, defined by the d.f. exp−1u\exp \left(-\frac{1}{u}\right)for u>0u\gt 0. The d.f. FFis in a one-to-one correspondence with its associated stable tail dependence function (“STDF”), defined by ℓ(x)≔−logF1x,x∈R+d,\ell \left(x):= -\log F\left(\frac{1}{x}\right),\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d},where 1x≔1x1,1x2,…\frac{1}{x}:= \left(\frac{1}{{x}_{1}},\frac{1}{{x}_{2}},\ldots \right), and these STDFs allow an intrinsic characterisation: ℓ:R+d→R\ell :{{\mathbb{R}}}_{+}^{d}\to {\mathbb{R}}is a STDF iff ℓ\ell is homogeneous (ℓ(tx)=t⋅ℓ(x)∀t,∀x\ell \left(tx)=t\cdot \ell \left(x)\hspace{1em}\forall t,\forall x), normalised (ℓ(ei)=1∀\ell \left({e}_{i})=1\hspace{1em}\forall unit vector ei{e}_{i}), and “fully dd-alternating” (to be explained later on), cf. [6], Theorem 6.The marginals of FFare given by Fα(xα)≔F(xα,∞αc)=exp−ℓ1xα,0αc{F}_{\alpha }\left({x}_{\alpha }):= F\left({x}_{\alpha },{\infty }_{{\alpha }^{c}})=\exp \left[-\ell \left(\frac{1}{{x}_{\alpha }},{{\bf{0}}}_{{\alpha }^{c}}\right)\right]for ∅≠α⊆{1,…,d},xα∈R+α\varnothing \ne \alpha \subseteq \left\{1,\ldots ,d\right\},{x}_{\alpha }\in {{\mathbb{R}}}_{+}^{\alpha }. Fα{F}_{\alpha }is again a MEV distribution with STDF ℓα≔ℓ∣R+α{\ell }_{\alpha }:= \ell | {{\mathbb{R}}}_{+}^{\alpha }. If X=(X1,…,Xd)X=\left({X}_{1},\ldots ,{X}_{d})has the d.f. FF, the subvector Xα≔(Xi,i∈α){X}_{\alpha }:= \left({X}_{i},i\in \alpha )has d.f. Fα{F}_{\alpha }.Two main subjects will be treated in this article. The first one is about the so-called extremal coefficients of a (dd-variate) STDF ℓ\ell , defined by ℓ(α)≔ℓ(1α),α⊆[d]\ell \left(\alpha ):= \ell \left({{\bf{1}}}_{\alpha }),\hspace{1.0em}\alpha \subseteq \left[d](slightly abusing notation). Although ℓ\ell is plainly not determined by its restriction to {0,1}d{\left\{0,1\right\}}^{d}, these coefficients contain important information, especially with respect to the independence of subvectors (Theorem 3).The other main theme addressed is about logistic, negative logistic and nested logistic STDFs. A certain functional equation (Theorem 5) turns out to be the key for several characterisations of “Archimedean type.” The well known sufficient conditions for “composebility” of logistic STDFs are shown to be necessary as well (Theorem 9) – meaning that the composed function is again a STDF.Except Theorem 1, all the other theorems in this article are new to the best of our knowledge. A recommendable treatment of STDFs is presented in chapter 8 of [1].Notations:R+≔[0,∞[{{\mathbb{R}}}_{+}:= {[}0,\infty {[}, N≔{1,2,3,…}{\mathbb{N}}:= \left\{1,2,3,\ldots \right\}, N0≔{0,1,2,…}{{\mathbb{N}}}_{0}:= \left\{0,1,2,\ldots \right\}, R¯≔[−∞,∞]\overline{{\mathbb{R}}}:= \left[-\infty ,\infty ],1x≔1x1,1x2,…\frac{1}{x}:= \left(\frac{1}{{x}_{1}},\frac{1}{{x}_{2}},\ldots \right)with 10≔∞,1∞≔0\frac{1}{0}:= \infty ,\frac{1}{\infty }:= 0[d]≔{1,…,d}\left[d]:= \left\{1,\ldots ,d\right\}, −α≔αc=[d]⧹α-\alpha := {\alpha }^{c}=\left[d]\setminus \alpha for a⊆[d]a\subseteq \left[d], 1d≔(1,…,1)∈Nd{{\bf{1}}}_{d}:= \left(1,\ldots ,1)\in {{\mathbb{N}}}^{d}, e1,…,ed{e}_{1},\ldots ,{e}_{d}are the usual unit vectors in Rd{{\mathbb{R}}}^{d}, 1α≔∑i∈αei{{\bf{1}}}_{\alpha }:= \sum _{i\in \alpha }{e}_{i}(f×g)(x,y)≔(f(x),g(y))(f\times g)\left(x,y):= (f\left(x),g(y))for mappings f,gf,gM+(X){M}_{+}\left(X)is the set of Radon measures on a locally compact space XXd.f. = distribution function.2Fully d-alternating functionsTo define this notion, which is of particular importance in this article, we introduce a special notation for multivariate real-valued functions. Let A1,…,Ad{A}_{1},\ldots ,{A}_{d}be non-empty sets, A≔A1×⋯×AdA:= {A}_{1}\times \cdots \times {A}_{d}, and f:A→Rf:A\to {\mathbb{R}}. First, for x∈Ax\in Aand ∅≠u⊆[d]\varnothing \ne u\subseteq \left[d], we put xu≔(xi)i∈u{x}_{u}:= {\left({x}_{i})}_{i\in u}, −u≔[d]⧹u-u:= \left[d]\setminus u, and so for x,z∈Ax,z\in A(zu,x−u)≔zi,i∈uxi,i∈−u,\left({z}_{u},{x}_{-u}):= \left\{\begin{array}{l}{z}_{i},i\in u\hspace{1.0em}\\ {x}_{i},i\in -u,\hspace{1.0em}\end{array}\right.i.e., another element of AA, being xxfor u=∅u=\varnothing and zzfor u=[d]u=\left[d]. Also, Au≔∏i∈uAi{A}_{u}:= {\prod }_{i\in u}{A}_{i}for u≠∅u\ne \varnothing . We then define Dzxf≔∑u⊆[d](−1)∣u∣f(zu,x−u)=f(x)∓⋯+(−1)df(z).{D}_{z}^{x}f:= \sum _{u\subseteq \left[d]}{\left(-1)}^{| u| }f\left({z}_{u},{x}_{-u})=f\left(x)\hspace{0.33em}\mp \cdots +\hspace{0.33em}{\left(-1)}^{d}f\left(z).Note that Dzxf=(−1)dDxzf{D}_{z}^{x}f={\left(-1)}^{d}{D}_{x}^{z}f. For ∅≠u⊊[d]\varnothing \ne u\hspace{0.33em}\subsetneq \hspace{0.33em}\left[d]and y−u∈A−u{y}_{-u}\in {A}_{-u}, we define a “partial version” of ffwith fixed values in the variables i∈−ui\in -uby f(⋅,y−u)(xu)≔f(xu,y−u),xu∈Au.f\left(\cdot ,{y}_{-u})\left({x}_{u}):= f\left({x}_{u},{y}_{-u}),\hspace{1.0em}{x}_{u}\in {A}_{u}.(For u=[d]u=\left[d], this would be ff, and for u=∅u=\varnothing , the constant f(y[d])f({y}_{\left[d]}).)There is a two-step procedure to determine Dzxf{D}_{z}^{x}fwhich will be needed later on:Lemma 1Let f:A→Rf:A\to {\mathbb{R}}, ∅≠v⊊[d]\varnothing \ne v\hspace{0.33em}\subsetneq \hspace{0.33em}\left[d], x,z∈Ax,z\in A, and define g:A−v→Rg:{A}_{-v}\to {\mathbb{R}}byg(y−v)≔Dzvxvf(⋅,y−v),y−v∈A−v.g({y}_{-v}):= {D}_{{z}_{v}}^{{x}_{v}}f\left(\cdot ,{y}_{-v}),\hspace{1.0em}{y}_{-v}\in {A}_{-v}.ThenDzxf=Dz−vx−vg.{D}_{z}^{x}f={D}_{{z}_{-v}}^{{x}_{-v}}g.ProofDz−vx−vg=∑w⊆−v(−1)∣w∣g(zw,x(−v)⧹w)=∑w⊆−v(−1)∣w∣∑u⊆v(−1)∣u∣f(zu,xv⧹u,zw,x(−v)⧹w)(notingu∩w=∅here, puttingα≔u∪w)=∑α⊆[d](−1)∣α∣f(zα,x−α)=Dzxf.□\hspace{12em}\begin{array}{rcl}{D}_{{z}_{-v}}^{{x}_{-v}}g& =& \displaystyle \sum _{w\subseteq -v}{\left(-1)}^{| w| }g\left({z}_{w},{x}_{\left(-v)\setminus w})\\ & =& \displaystyle \sum _{w\subseteq -v}{\left(-1)}^{| w| }\displaystyle \sum _{u\subseteq v}{\left(-1)}^{| u| }f\left({z}_{u},{x}_{v\setminus u},{z}_{w},{x}_{\left(-v)\setminus w})\\ & & \left(\hspace{0.1em}\text{noting}\hspace{0.1em}\hspace{0.33em}u\cap w=\varnothing \hspace{0.33em}\hspace{0.1em}\text{here, putting}\hspace{0.1em}\hspace{0.33em}\alpha := u\cup w)\\ & =& \displaystyle \sum _{\alpha \subseteq \left[d]}{\left(-1)}^{| \alpha | }f\left({z}_{\alpha },{x}_{-\alpha })\\ & =& {D}_{z}^{x}f.\hspace{28em}\square \end{array}DefinitionLet A1,…,Ad⊆R¯{A}_{1},\ldots ,{A}_{d}\subseteq \overline{{\mathbb{R}}}be non-empty, A≔A1×⋯×AdA:= {A}_{1}\times \cdots \times {A}_{d}. Then f:A→Rf:A\to {\mathbb{R}}is fully d-alternating (in symbols “1d-↕{{\bf{1}}}_{d}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow ”) iff Dzxf≤0{D}_{z}^{x}f\le 0for x≤zx\le z(both in AA), and if also Dzvxvf(⋅,y−v)≤0{D}_{{z}_{v}}^{{x}_{v}}f\left(\cdot ,{y}_{-v})\le 0for each ∅≠v⊊[d]\varnothing \ne v\hspace{0.33em}\subsetneq \hspace{0.33em}\left[d]and xv≤zv{x}_{v}\le {z}_{v}(both in Av{A}_{v}), and for all y−v∈A−v{y}_{-v}\in {A}_{-v}.This property is specific for co-survival functions, i.e., f(x)≔P(X≱x)f\left(x):= P\left(X\ge &#x0338;x), e.g., for XXuniform on [0,1]d{\left[0,1]}^{d}f(x)=∑xi−∑i<jxixj+∑i<j<kxixjxk∓⋯,f\left(x)=\sum {x}_{i}-\sum _{i\lt j}{x}_{i}{x}_{j}+\sum _{i\lt j\lt k}{x}_{i}{x}_{j}{x}_{k}\mp \cdots \hspace{0.33em},but it is of special importance also for some infinite measures, as we will see shortly.Remark 1There is a more general notion of n{\bf{n}}-alternating (“n-↕{\bf{n}}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow ”) functions, with n∈Nd{\bf{n}}\in {{\mathbb{N}}}^{d}, cf. [7], describing monotonicity conditions of higher orders, not needed in this work.For A1=⋯=Ad=R+{A}_{1}=\cdots ={A}_{d}={{\mathbb{R}}}_{+}, i.e., A=R+dA={{\mathbb{R}}}_{+}^{d}, a function ℓ:A→R\ell :A\to {\mathbb{R}}is a STDF iff ℓ\ell is homogeneous (i.e., ℓ(tx)=t⋅ℓ(x)∀t≥0,∀x∈R+d\ell \left(tx)=t\cdot \ell \left(x)\hspace{1em}\forall t\ge 0,\hspace{1em}\forall x\in {{\mathbb{R}}}_{+}^{d}), 1d-↕{{\bf{1}}}_{d}\hspace{-0.33em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.3em}\updownarrow and normalised, i.e., ℓ(ei)=1\ell \left({e}_{i})=1for each unit vector. Disregarding normalisation, we consider K≔{f:R+d→R+∣fis1d-↕and homogeneous,f(1d)=1}.K:= \{f:{{\mathbb{R}}}_{+}^{d}\to {{\mathbb{R}}}_{+}| f\hspace{0.33em}\hspace{0.1em}\text{is}\hspace{0.1em}\hspace{0.33em}{{\bf{1}}}_{d}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow \hspace{0.33em}\hspace{0.1em}\text{and homogeneous}\hspace{0.1em},\hspace{0.33em}f\left({{\bf{1}}}_{d})=1\}.This set, obviously compact and convex, was shown in ref. [6] to be a so-called Bauer simplex (i.e., a compact convex subset of some locally convex Hausdorff space, for which the extreme boundary is closed, and for which the integral representation given by the Krein-Milman theorem is unique), with extreme boundary ex(K)={x↦maxi≤d(xiwi)∣w∈Cd},{\rm{ex}}\left(K)=\left\{\phantom{\rule[-1.25em]{}{0ex}}x\mapsto \mathop{\max }\limits_{i\le d}\left({x}_{i}{w}_{i})| w\in {C}_{d}\right\},where Cd≔{w∈[0,1]d∣maxi≤dwi=1}{C}_{d}:= \left\{w\in {\left[0,1]}^{d}| {\max }_{i\le d}{w}_{i}=1\right\}, and for each homogeneous 1d-↕{{\bf{1}}}_{d}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow function ff(≢0\not\equiv 0), we have the unique integral representation f(x)=f(1d)⋅∫maxi≤d(xiwi)dν(w)f\left(x)=f\left({{\bf{1}}}_{d})\cdot \int \mathop{\max }\limits_{i\le d}\left({x}_{i}{w}_{i}){\rm{d}}\nu \left(w)with ν\nu a probability measure on Cd{C}_{d}.The function ffis then the so-called co-survival function of a homogeneous Radon measure μ\mu on the locally compact space Zd≔[0,∞]d⧹{∞d}{Z}_{d}:= {\left[0,\infty ]}^{d}\setminus \left\{{\infty }_{d}\right\}, i.e., f(x)=μ([x,∞]c)≕μˇ(x)f\left(x)=\mu \left({\left[x,\infty ]}^{c})\hspace{0.33em}=: \hspace{0.33em}\check{\mu }\left(x)(which is finite by the definition of a Radon measure).3Properties depending on the extremal coefficientsLet ℓ\ell be a dd-variate STDF. Its restriction ℓ∣{0,1}d\ell | {\left\{0,1\right\}}^{d}gives the so-called extremal coefficients ℓ(α)≔ℓ(1α)\ell \left(\alpha ):= \ell \left({{\bf{1}}}_{\alpha })for α⊆[d]\alpha \subseteq \left[d](hence, ℓ([d])=ℓ(1d)\ell \left(\left[d])=\ell \left({{\bf{1}}}_{d})). From the integral representation, ℓ(x)=ℓ(1d)⋅∫Cdmaxi≤d(xiwi)dν(w),\ell \left(x)=\ell \left({{\bf{1}}}_{d})\cdot \mathop{\int }\limits_{{C}_{d}}\mathop{\max }\limits_{i\le d}\left({x}_{i}{w}_{i}){\rm{d}}\nu \left(w),we obtain ℓ(α)=ℓ(1d)⋅∫maxi∈αwidν(w),∅≠α⊆[d].\ell \left(\alpha )=\ell \left({{\bf{1}}}_{d})\cdot \int \mathop{\max }\limits_{i\in \alpha }{w}_{i}{\rm{d}}\nu \left(w),\hspace{1.0em}\varnothing \ne \alpha \subseteq \left[d].Clearly, the restriction of ℓ\ell to {0,1}d{\left\{0,1\right\}}^{d}does in general not determine ℓ\ell , with the following exception, known since a long time, and due to Takahashi [9,10]. Note that maxi≤dxi≤ℓ(x)≤∑i=1dxi\mathop{\max }\limits_{i\le d}{x}_{i}\le \ell \left(x)\le \mathop{\sum }\limits_{i=1}^{d}{x}_{i}holds for any dd-variate STDF ℓ\ell , and therefore, 1≤ℓ(1d)≤d.1\le \ell \left({{\bf{1}}}_{d})\le d.Theorem 1Let ℓ\ell be a dd-variate STDF. Then, (i)ℓ(1d)=1⇔ℓ(x)=maxi≤dxi∀x\ell \left({{\bf{1}}}_{d})=1\iff \ell \left(x)={\max }_{i\le d}{x}_{i}\hspace{1.0em}\forall x(ii)ℓ(1d)=d⇔ℓ(x)=∑i=1dxi∀x\ell \left({{\bf{1}}}_{d})=d\iff \ell \left(x)={\sum }_{i=1}^{d}{x}_{i}\hspace{1.0em}\forall x.Proof(i)If ℓ(1d)=1\ell \left({{\bf{1}}}_{d})=1, then ℓ(ei)=∫widν(w)=1\ell \left({e}_{i})=\int {w}_{i}{\rm{d}}\nu \left(w)=1; hence, ν({w∈Cd∣wi=1})=1∀i≤d\nu \left(\left\{w\in {C}_{d}| {w}_{i}=1\right\})=1\hspace{1em}\forall i\le d, i.e., ν({1d})=1\nu \left(\left\{{{\bf{1}}}_{d}\right\})=1and ℓ(x)=\ell \left(x)=maxi≤dxi{\max }_{i\le d}{x}_{i}.(ii)If ℓ(1d)=d\ell \left({{\bf{1}}}_{d})=d, then ∫widν(w)=1d\int {w}_{i}{\rm{d}}\nu \left(w)=\frac{1}{d}for each ii; hence, ∫∑i=1dwidν(w)=1=∫maxi≤dwidν(w),\int \mathop{\sum }\limits_{i=1}^{d}{w}_{i}{\rm{d}}\nu \left(w)=1=\int \mathop{\max }\limits_{i\le d}{w}_{i}{\rm{d}}\nu \left(w),i.e., ∑i=1dwi=maxi≤dwi=1ν{\sum }_{i=1}^{d}{w}_{i}={\max }_{i\le d}{w}_{i}=1\hspace{0.33em}\nu -a.s., or ν({e1,…,ed})=1\nu \left(\left\{{e}_{1},\ldots ,{e}_{d}\right\})=1. From 1d=∫widν(w)=ν({ei})∀i≤d,\frac{1}{d}=\int {w}_{i}{\rm{d}}\nu \left(w)=\nu \left(\left\{{e}_{i}\right\})\hspace{1.0em}\forall i\le d,we deduce ν=1d∑i=1dεei\nu =\frac{1}{d}{\sum }_{i=1}^{d}{\varepsilon }_{{e}_{i}}, or ℓ(x)=∑i=1dxi\ell \left(x)={\sum }_{i=1}^{d}{x}_{i}.□Let X=(X1,…,Xd)X=\left({X}_{1},\ldots ,{X}_{d})have the MEV-distribution associated with the STDF ℓ\ell , i.e., with d.f. F(x)=exp−ℓ1xF\left(x)=\exp \left[-\ell \left(\frac{1}{x}\right)\right], for x∈]0,∞]dx\in {]0,\infty ]}^{d}. For ∅≠α⊆[d]\varnothing \ne \alpha \subseteq \left[d], the subvector Xα≔(Xi,i∈α){X}_{\alpha }:= \left({X}_{i},i\in \alpha )then has the d.f.Fα(xα)=F(xα,∞−α)=exp[−ℓ(xα−1,0−α)],{F}_{\alpha }\left({x}_{\alpha })=F\left({x}_{\alpha },{\infty }_{-\alpha })=\exp {[}-\ell ({x}_{\alpha }^{-1},{{\bf{0}}}_{-\alpha })],including F{i}(xi)=exp−1xi{F}_{\left\{i\right\}}\left({x}_{i})=\exp \left(-\frac{1}{{x}_{i}}\right), i=1,…,di=1,\ldots ,d.Condition (i) in Theorem 1 means that a.s. X1=X2=⋯=Xd{X}_{1}={X}_{2}=\cdots ={X}_{d}, and (ii) is equivalent with X1,…,Xd{X}_{1},\ldots ,{X}_{d}being iid (standard Fréchet). The independence of two subvectors of XXalso depends only on the extremal coefficients, as we now shall see.Theorem 2For disjoint (non-empty) subsets α,β⊆[d]\alpha ,\beta \subseteq \left[d], the following properties are equivalent: (i)ℓ(α)+ℓ(β)=ℓ(α∪β)\ell \left(\alpha )+\ell \left(\beta )=\ell \left(\alpha \cup \beta )(ii)ℓ(xα,0−α)+ℓ(xβ,0−β)=ℓ(xα∪β,0−(α∪β))∀x\ell ({x}_{\alpha },{{\bf{0}}}_{-\alpha })+\ell ({x}_{\beta },{{\bf{0}}}_{-\beta })=\ell ({x}_{\alpha \cup \beta },{{\bf{0}}}_{-\left(\alpha \cup \beta )})\hspace{1em}\forall x(iii)Xα{X}_{\alpha }and Xβ{X}_{\beta }are independent.ProofIn view of the connection between ℓ\ell and FF, only (i) ⇒\Rightarrow (ii) has to be shown. Without restriction α∪β=[d]\alpha \cup \beta =\left[d], i.e., β=−α\beta =-\alpha . So, let us assume (i), then from ℓ(α)=ℓ(1d)∫maxi∈αwidν(w)ℓ(β)=ℓ(1d)∫maxi∈βwidν(w),\begin{array}{rcl}\ell \left(\alpha )& =& \ell \left({{\bf{1}}}_{d})\displaystyle \int \mathop{\max }\limits_{i\in \alpha }{w}_{i}{\rm{d}}\nu \left(w)\\ \ell \left(\beta )& =& \ell \left({{\bf{1}}}_{d})\displaystyle \int \mathop{\max }\limits_{i\in \beta }{w}_{i}{\rm{d}}\nu \left(w),\end{array}we obtain ∫(maxi∈αwi+maxi∈βwi)dν(w)=∫maxi≤dwidν(w)=1.\int \left(\mathop{\max }\limits_{i\in \alpha }{w}_{i}+\mathop{\max }\limits_{i\in \beta }{w}_{i}\right){\rm{d}}\nu \left(w)=\int \mathop{\max }\limits_{i\le d}{w}_{i}{\rm{d}}\nu \left(w)=1.Let f(w)≔maxi∈αwif\left(w):= {\max }_{i\in \alpha }{w}_{i}, g(w)≔maxi∈βwig\left(w):= {\max }_{i\in \beta }{w}_{i}, w∈Cdw\in {C}_{d}. Then 0≤f≤10\le f\le 1, 0≤g≤10\le g\le 1, f∨g=1f\vee g=1, ∫(f+g)dν=∫f∨gdν\int (f+g){\rm{d}}\nu =\int f\vee g{\rm{d}}\nu . Since f∨g+f∧g=f+gf\vee g+f\wedge g=f+g, we obtain ∫f∧gdν=0\int f\wedge g{\rm{d}}\nu =0, or ν({f>0}∩{g>0})=0\nu (\{f\gt 0\}\cap \left\{g\gt 0\right\})=0, and Cd={f=1}∪{g=1}{C}_{d}=\{f=1\}\cup \left\{g=1\right\}. It follows ℓ(x)=ℓ(1d)∫maxi≤d(xiwi)dν(w)=ℓ(1d)⋅∫{f=1}…+∫{g=1}…=ℓ(xα,0−α)+ℓ(xβ,0−β).□\hspace{5em}\ell \left(x)=\ell \left({{\bf{1}}}_{d})\int \mathop{\max }\limits_{i\le d}\left({x}_{i}{w}_{i}){\rm{d}}\nu \left(w)=\ell \left({{\bf{1}}}_{d})\cdot \left(\mathop{\int }\limits_{\{f=1\}}\ldots +\mathop{\int }\limits_{\left\{g=1\right\}}\ldots \right)=\ell \left({x}_{\alpha },{{\bf{0}}}_{-\alpha })+\ell \left({x}_{\beta },{{\bf{0}}}_{-\beta }).\hspace{5em}\square Before we extend Theorem 2 to more than two subvectors, we need the following.Lemma 2Let I⊆R¯I\subseteq \overline{{\mathbb{R}}}be any non-degenerate interval, f:Id→Ra1d-↕f:{I}^{d}\to {\mathbb{R}}\hspace{0.33em}a\hspace{0.33em}{{\bf{1}}}_{d}\hspace{-0.33em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.3em}\updownarrow function, and [d]=α1∪α2∪⋯∪αn\left[d]={\alpha }_{1}\cup {\alpha }_{2}\cup \cdots \cup {\alpha }_{n}a partition with non-empty α1,…,αn{\alpha }_{1},\ldots ,{\alpha }_{n}. Define g:In→Rg:{I}^{n}\to {\mathbb{R}}by g(x1,…,xn)≔f∑j=1nxj1αjg\left({x}_{1},\ldots ,{x}_{n}):= f\left({\sum }_{j=1}^{n}{x}_{j}{{\bf{1}}}_{{\alpha }_{j}}\right). Then ggis 1n-↕{{\bf{1}}}_{n}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow . If ffis homogeneous, so is gg.Proofd=2d=2being trivial, assume d≥3d\ge 3. It is clearly enough to consider the partition α1={1},…,αd−2={d−2},αd−1={d−1,d},{\alpha }_{1}=\left\{1\right\},\ldots ,{\alpha }_{d-2}=\left\{d-2\right\},{\alpha }_{d-1}=\left\{d-1,d\right\},since the general case then follows easily by iteration.We have g(x1,…,xd−1)=f(x1,…,xd−2,xd−1,xd−1)g\left({x}_{1},\ldots ,{x}_{d-1})=f\left({x}_{1},\ldots ,{x}_{d-2},{x}_{d-1},{x}_{d-1}). Let x,z∈Id−1x,z\in {I}^{d-1}with x≤zx\le z, define x′≔(x1,…,xd−2)x^{\prime} := \left({x}_{1},\ldots ,{x}_{d-2}), and z′≔(z1,…,zd−2)z^{\prime} := \left({z}_{1},\ldots ,{z}_{d-2}), then by Lemma 1, Dzxg=Dz′x′g(⋯,xd−1)−Dz′x′g(⋯,zd−1)=Dz′x′f(⋯,xd−1,xd−1)−Dz′x′f(⋯,zd−1,zd−1)=[Dz′x′f(⋯,xd−1,xd−1)−Dz′x′f(⋯,zd−1,xd−1)]+[Dz′x′f(⋯,zd−1,xd−1)−Dz′x′f(⋯,zd−1,zd−1)]=Dzxf(⋯,⋅,xd−1)+Dzxf(⋯,zd−1,⋅)≤0\begin{array}{rcl}{D}_{z}^{x}g& =& {D}_{z^{\prime} }^{x^{\prime} }g\left(\cdots \hspace{0.33em},{x}_{d-1})-{D}_{z^{\prime} }^{x^{\prime} }g\left(\cdots \hspace{0.33em},{z}_{d-1})\\ & =& {D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{x}_{d-1},{x}_{d-1})-{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{z}_{d-1},{z}_{d-1})\\ & =& {[}{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{x}_{d-1},{x}_{d-1})-{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{z}_{d-1},{x}_{d-1})]+{[}{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{z}_{d-1},{x}_{d-1})-{D}_{z^{\prime} }^{x^{\prime} }f\left(\cdots \hspace{0.33em},{z}_{d-1},{z}_{d-1})]\\ & =& {D}_{z}^{x}f\left(\cdots \hspace{0.33em},\cdot ,{x}_{d-1})+{D}_{z}^{x}f\left(\cdots \hspace{0.33em},{z}_{d-1},\cdot )\\ & \le & 0\end{array}as the sum of two non-positive numbers.□The following result is a considerable generalisation of Theorem 1 (ii) and Theorem 2.Theorem 3Let ℓ\ell be a dd-variate STDF, and let α1,…,αn{\alpha }_{1},\ldots ,{\alpha }_{n}be disjoint non-empty subsets of [d]\left[d]. The random vector X=(X1,…,Xd)X=\left({X}_{1},\ldots ,{X}_{d})is supposed to have the d.f. exp−ℓ1x\exp \left[-\ell \left(\frac{1}{x}\right)\right], x∈R+dx\in {{\mathbb{R}}}_{+}^{d}. Then the following conditions are equivalent: (i)ℓ(⋃j=1nαj)=∑j=1nℓ(αj)\ell ({\bigcup }_{j=1}^{n}{\alpha }_{j})={\sum }_{j=1}^{n}\ell \left({\alpha }_{j}).(ii){Xαj∣j=1,…,n}\left\{{X}_{{\alpha }_{j}}| j=1,\ldots ,n\right\}are independent.(iii){Xαj∣j=1,…,n}\left\{{X}_{{\alpha }_{j}}| j=1,\ldots ,n\right\}are pairwise independent.ProofWithout restriction, we assume ⋃j=1nαj=[d]{\bigcup }_{j=1}^{n}{\alpha }_{j}=\left[d].(i) ⇒\Rightarrow (ii): We use induction, the case n=2n=2being true by Theorem 2. Supposing the conclusion for nn, we use ℓ\ell ’s subadditivity to obtain ℓ⋃j=1n+1αj≤ℓ⋃j=1nαj+ℓ(αn+1)≤∑j=1nℓ(αj)+ℓ(αn+1)=ℓ⋃j=1n+1αj,\ell \left(\mathop{\bigcup }\limits_{j=1}^{n+1}{\alpha }_{j}\right)\le \ell \left(\mathop{\bigcup }\limits_{j=1}^{n}{\alpha }_{j}\right)+\ell \left({\alpha }_{n+1})\le \mathop{\sum }\limits_{j=1}^{n}\ell \left({\alpha }_{j})+\ell \left({\alpha }_{n+1})=\ell \left(\mathop{\bigcup }\limits_{j=1}^{n+1}{\alpha }_{j}\right),and hence, ℓ(⋃j=1nαj)=∑j=1nℓ(αj)\ell ({\bigcup }_{j=1}^{n}{\alpha }_{j})={\sum }_{j=1}^{n}\ell \left({\alpha }_{j}), {Xαj∣j≤n}\left\{{X}_{{\alpha }_{j}}| j\le n\right\}are independent, and n=2n=2may be applied to ⋃j=1nαj{\bigcup }_{j=1}^{n}{\alpha }_{j}and αn+1{\alpha }_{n+1}.(iii) ⇒\Rightarrow (i): We use again induction. For n=2n=2, there is nothing to prove. We assume validity for some n≥2n\ge 2and consider the case n+1n+1. Let α≔⋃j<nαj,β≔αn,γ≔αn+1\alpha := \bigcup _{j\lt n}{\alpha }_{j},\hspace{1.0em}\beta := {\alpha }_{n},\hspace{1.0em}\gamma := {\alpha }_{n+1}and define ffon R+3{{\mathbb{R}}}_{+}^{3}by f(a,b,c)≔ℓ(a1α+b1β+c1γ)f\left(a,b,c):= \ell \left(a{{\bf{1}}}_{\alpha }+b{{\bf{1}}}_{\beta }+c{{\bf{1}}}_{\gamma }). By Lemma 2, ffis 13-↕{{\bf{1}}}_{3}\hspace{-0.25em}\hspace{0.1em}\text{-}\hspace{0.1em}\hspace{-0.1em}\updownarrow (not normalised!). Therefore, 0≥D1303f=f(0)−∑f(ei)+∑i<jf(ei+ej)−f(13)=0−ℓ(α)−ℓ(β)−ℓ(γ)+ℓ(α∪β)+ℓ(α∪γ)+ℓ(β∪γ)−ℓ(α∪β∪γ)=−∑j<nℓ(αj)−ℓ(αn)−ℓ(αn+1)+∑j≤nℓ(αj)+∑j≠nℓ(αj)+(ℓ(αn)+ℓ(αn+1))−ℓ⋃j=1n+1αj,\begin{array}{rcl}0& \ge & {D}_{{{\bf{1}}}_{3}}^{{{\bf{0}}}_{3}}f=f\left(0)-\displaystyle \sum f\left({e}_{i})+\displaystyle \sum _{i\lt j}f\left({e}_{i}+{e}_{j})-f\left({{\bf{1}}}_{3})\\ & =& 0-\ell \left(\alpha )-\ell \left(\beta )-\ell \left(\gamma )+\ell \left(\alpha \cup \beta )+\ell \left(\alpha \cup \gamma )+\ell \left(\beta \cup \gamma )-\ell \left(\alpha \cup \beta \cup \gamma )\\ & =& -\displaystyle \sum _{j\lt n}\ell \left({\alpha }_{j})-\ell \left({\alpha }_{n})-\ell \left({\alpha }_{n+1})+\displaystyle \sum _{j\le n}\ell \left({\alpha }_{j})+\displaystyle \sum _{j\ne n}\ell \left({\alpha }_{j})+\left(\ell \left({\alpha }_{n})+\ell \left({\alpha }_{n+1}))-\ell \left(\mathop{\bigcup }\limits_{j=1}^{n+1}{\alpha }_{j}\right),\end{array}i.e., ∑j=1n+1ℓ(αj)≤ℓ⋃j=1n+1αj≤∑j=1n+1ℓ(αj).□\hspace{16em}\mathop{\sum }\limits_{j=1}^{n+1}\ell \left({\alpha }_{j})\le \ell \left(\mathop{\bigcup }\limits_{j=1}^{n+1}{\alpha }_{j}\right)\le \mathop{\sum }\limits_{j=1}^{n+1}\ell \left({\alpha }_{j}).\hspace{16.5em}\square Considering in Theorem 3 the special case αj={j}{\alpha }_{j}=\left\{j\right\}, j=1,…,dj=1,\ldots ,d, we are back to Theorem 1 (ii), with the additional equivalence to pairwise independence, i.e., ℓ(ei+ej)=2∀i≠j\ell \left({e}_{i}+{e}_{j})=2\hspace{1em}\forall i\ne j. One might be tempted to believe that there is a corresponding generalisation of part (i) of Theorem 1 as well. This is not the case.Theorem 4(i)Let ℓ\ell be a d-variate STDF; α,β⊆[d]\alpha ,\beta \subseteq \left[d]such that α∩β≠∅\alpha \cap \beta \ne \varnothing . If ℓ(α)=ℓ(β)=1\ell \left(\alpha )=\ell \left(\beta )=1, then also ℓ(α∪β)=1\ell \left(\alpha \cup \beta )=1.(ii)Let f,gf,gbe mm- (resp. nn-)variate STDFs, such that also ℓ(x,y)≔f(x)∨g(y)\ell \left(x,y):= f\left(x)\vee g(y)is a STDF. Then f=maxf=\max , g=maxg=\max (and ℓ=max\ell =\max ).Proof(i)Let XXbe a R+d{{\mathbb{R}}}_{+}^{d}-valued random vector with STDF ℓ\ell . Then, if ℓ(α)=ℓ(β)=1,Xi=Xj\ell \left(\alpha )=\ell \left(\beta )=1,{X}_{i}={X}_{j}a.s. ∀i,j∈α\forall \hspace{-0.3em}i,j\in \alpha and ∀i,j∈β\forall \hspace{-0.3em}i,j\in \beta , and because of α∩β≠∅\alpha \cap \beta \ne \varnothing , Xi=Xj{X}_{i}={X}_{j}a.s. ∀i,j∈α∪β\forall \hspace{-0.3em}i,j\in \alpha \cup \beta . That is, ℓ(α∪β)=1\ell \left(\alpha \cup \beta )=1.(ii)Again let (X1,…,Xm,Y1,…,Yn)\left({X}_{1},\ldots ,{X}_{m},{Y}_{1},\ldots ,{Y}_{n})have ℓ\ell as its STDF. Then, Xi=Yj{X}_{i}={Y}_{j}a.s. ∀i∈[m]\forall \hspace{-0.3em}i\in \left[m], ∀j∈[n]\forall \hspace{-0.3em}j\in \left[n], i.e., X1=⋯=Xm=Y1=⋯=Yn{X}_{1}=\cdots ={X}_{m}={Y}_{1}=\cdots ={Y}_{n}a.s., leading to ℓ=max\ell =\max , f=maxf=\max and g=maxg=\max .□Remark 2For “overlapping variables”, this is different: ℓ(x,y,z)≔(x+y)∨(y+z)=x∨z+y\ell \left(x,y,z):= \left(x+y)\vee (y+z)=x\vee z+yis a STDF, as is also (with a,b∈[0,1]a,b\in \left[0,1]) ℓ(x,y)≔(ax+y)∨(x+by)=ax+by+[(1−a)x]∨[(1−b)y].\ell \left(x,y):= \left(ax+y)\vee \left(x+by)=ax+by+\left[\left(1-a)x]\vee \left[\left(1-b)y].With iid standard Fréchet random variables XX, YY, and ZZ, a stochastic model for these two STDFs would be the random vector (X,Y,X)\left(X,Y,X), resp. ((aX)∨(1−a)Z,(bY)∨(1−b)Z)\left(\left(aX)\vee \left(1-a)Z,\left(bY)\vee \left(1-b)Z).Note, however, that f(x,y,z)≔(x+y)∨(y+z)∨(z+x)f\left(x,y,z):= \left(x+y)\vee (y+z)\vee \left(z+x)is not a STDF: D1303f=0−3+6−2=1>0{D}_{{{\bf{1}}}_{3}}^{{{\bf{0}}}_{3}}f=0-3+6-2=1\gt 0.4Characterisation of logistic and related STDFsPerhaps the best-known STDFs are the logistic ones, i.e., the family {ℓp∣p∈[1,∞[}\left\{{\ell }_{p}| p\in {[}1,\infty {[}\right\}, defined by ℓp(x)≔∑i=1dxip1/p,x∈R+d.{\ell }_{p}\left(x):= {\left(\mathop{\sum }\limits_{i=1}^{d}{x}_{i}^{p}\right)}^{1\text{/}p},\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d}.Among all symmetric STDFs they are particular, depending on xxin an “additive way,” being a function of ∑i=1dg(xi){\sum }_{i=1}^{d}g\left({x}_{i})for some g:R+→R+g:{{\mathbb{R}}}_{+}\to {{\mathbb{R}}}_{+}. We shall see that there are no other STDFs with this property besides the logistic ones.We begin by solving a functional equation.Theorem 5Let φ:R+2→R+\varphi :{{\mathbb{R}}}_{+}^{2}\to {{\mathbb{R}}}_{+}be homogeneous, ≢0\not\equiv 0, and let g:[0,∞]→[0,∞]g:\left[0,\infty ]\to \left[0,\infty ]be a continuous bijection, such that g(1)=1g\left(1)=1andg(φ(x,y))=g(x)+g(y)∀x,y∈R+.g\left(\varphi \left(x,y))=g\left(x)+g(y)\hspace{1.0em}\forall x,y\in {{\mathbb{R}}}_{+}.Then, ∃p∈R⧹{0}\exists \hspace{-0.16em}p\in {\mathbb{R}}\setminus \left\{0\right\}such that g(x)=xp∀x∈]0,∞[g\left(x)={x}^{p}\hspace{1em}\forall x\in ]0,\infty {[}(which of course extends uniquely to [0,∞]\left[0,\infty ]).ProofObviously g(]0,∞[)=]0,∞[g\left(]0,\infty {[})=]0,\infty {[}and g({0,∞})={0,∞}g\left(\left\{0,\infty \right\})=\left\{0,\infty \right\}, and ggis either (strictly) increasing or decreasing. Since g−1{g}^{-1}is also continuous, so is φ\varphi .For a≔φ(1,1)a:= \varphi \left(1,1), we have g(a)=2g\left(a)=2and g(ta)=g(t⋅φ(1,1))=g(φ(t,t))=2g(t)∀t∈R+g\left(ta)=g\left(t\cdot \varphi \left(1,1))=g\left(\varphi \left(t,t))=2g\left(t)\hspace{1em}\forall t\in {{\mathbb{R}}}_{+}(i.e., g(0)=2g(0)g\left(0)=2g\left(0), in accordance with g(0)∈{0,∞}g\left(0)\in \left\{0,\infty \right\}). The equality g(ta)=g(t)g(a)g\left(ta)=g\left(t)g\left(a)shows aato belong to G≔{x∈]0,∞[∣g(tx)=g(t)g(x)∀t∈]0,∞[},G:= \left\{x\in ]0,\infty {[}| g\left(tx)=g\left(t)g\left(x)\hspace{1em}\forall t\in ]0,\infty {[}\right\},a multiplicative subgroup of ]0,∞[]0,\infty {[}as is easily seen. Hence, {an∣n∈Z}⊆G\left\{{a}^{n}| n\in {\mathbb{Z}}\right\}\subseteq G.For n∈Zn\in {\mathbb{Z}}, g(φ(1,an))=1+g(an)=1+(g(a))n=1+2n,g\left(\varphi \left(1,{a}^{n}))=1+g\left({a}^{n})=1+{\left(g\left(a))}^{n}=1+{2}^{n},and for t>0t\gt 0, g(t⋅φ(1,an))=g(φ(t,tan))=g(t)+g(tan)=g(t)(1+2n)=g(t)g(φ(1,an)),g\left(t\cdot \varphi \left(1,{a}^{n}))=g\left(\varphi \left(t,t{a}^{n}))=g\left(t)+g\left(t{a}^{n})=g\left(t)\left(1+{2}^{n})=g\left(t)g\left(\varphi \left(1,{a}^{n})),i.e., also {φ(1,an)∣n∈Z}⊆G\left\{\varphi \left(1,{a}^{n})| n\in {\mathbb{Z}}\right\}\subseteq G, where φ(1,an)=g−1(1+2n)\varphi \left(1,{a}^{n})={g}^{-1}\left(1+{2}^{n}), and this converges to g−1(1)=1{g}^{-1}\left(1)=1for n→−∞n\to -\infty .This implies GGto be dense in ]0,∞[]0,\infty {[}: it suffices to show 1<u<v⇒G∩]u,v[≠∅,1\lt u\lt v\Rightarrow G\cap ]u,v{[}\hspace{0.33em}\ne \hspace{0.33em}\varnothing ,and this follows because for any x∈1,vux\in \left]1,\frac{v}{u}\right[, {xj∣j∈N}∩]u,v[≠∅\left\{{x}^{j}| j\in {\mathbb{N}}\right\}\cap ]u,v{[}\hspace{0.33em}\ne \hspace{0.33em}\varnothing (choose kkwith xk−1≤u<xk{x}^{k-1}\le u\lt {x}^{k}, then xk=x⋅xk−1<vu⋅u=v{x}^{k}=x\cdot {x}^{k-1}\lt \frac{v}{u}\cdot u=v). If ggis increasing, then g−1(1+2n)∈1,vu{g}^{-1}\left(1+{2}^{n})\in \left]1,\frac{v}{u}\right[for some (negative!) nn, and for decreasing gg, we may choose instead [g−1(1+2n)]−1{\left[{g}^{-1}\left(1+{2}^{n})]}^{-1}.Now GGis closed, ggbeing continuous; hence, G=]0,∞[G=]0,\infty {[}and g(xy)=g(x)g(y)∀x,y∈]0,∞[g\left(xy)=g\left(x)g(y)\hspace{1em}\forall x,y\in ]0,\infty {[}. It is well known that this implies g(x)=xpg\left(x)={x}^{p}for some p≠0p\ne 0. (For f≔log∘g∘exp:R→Rf:= \log \circ g\circ \exp :{\mathbb{R}}\to {\mathbb{R}}, we have f(s+t)=f(s)+f(t)∀s,tf\left(s+t)=f\left(s)+f\left(t)\hspace{1em}\forall s,t; this is the standard Cauchy equation, and ffbeing continuous, it has the form f(s)=c⋅sf\left(s)=c\cdot swith c∈Rc\in {\mathbb{R}}; therefore, g(x)=xcg\left(x)={x}^{c}.) From g(a)=2=apg\left(a)=2={a}^{p}, we obtain p=log2loga>0fora>1<0fora<1.p=\frac{\log 2}{\log a}\left\{\begin{array}{l}\gt 0\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}a\gt 1\hspace{1.0em}\\ \lt 0\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}a\lt 1.\hspace{1.0em}\end{array}\right.□Theorem 6(Characterisation of logistic STDFs) Let ℓ\ell be a d-variate STDF of the formℓ(x)=g−1∑i=1dg(xi)\ell \left(x)={g}^{-1}\left(\mathop{\sum }\limits_{i=1}^{d}g\left({x}_{i})\right)for some continuous bijection g:R+→R+g:{{\mathbb{R}}}_{+}\to {{\mathbb{R}}}_{+}, g(1)=1g\left(1)=1without restriction.Then, g(x)=xpg\left(x)={x}^{p}for some p≥1p\ge 1, i.e., ℓ=ℓp\ell ={\ell }_{p}.ProofObviously ggis (strictly) increasing, in particular g(0)=0g\left(0)=0, and it suffices to consider d=2d=2. By the preceding theorem, g(x)=xpg\left(x)={x}^{p}for some p∈]0,∞[p\in ]0,\infty {[}, and g(ℓ(1,1))=(ℓ(1,1))p=2g\left(\ell \left(1,1))={\left(\ell \left(1,1))}^{p}=2implies ℓ(1,1)>1\ell \left(1,1)\gt 1, and p=log2logℓ(1,1)≥1p=\frac{\log 2}{\log \ell \left(1,1)}\ge 1since ℓ(1,1)≤2\ell \left(1,1)\le 2.□This result, assuming from the outset (though tacitly) the function ggto be differentiable, was shown in an equivalent form for copulas, stating that the only Archimedean extreme value copulas are the logistic (or Gumbel) ones, cf. [3]. We state this as a corollary, being slightly more general while not assuming differentiability:Corollary 1Let CCbe a dd-variate Archimedean copula, i.e., C(u)=h−1∑i=1dh(ui),u∈]0,1]dC\left(u)={h}^{-1}\left(\mathop{\sum }\limits_{i=1}^{d}h\left({u}_{i})\right),\hspace{1.0em}u\in {]0,1]}^{d}with a decreasing bijection h:]0,1]→R+h:]0,1]\to {{\mathbb{R}}}_{+}, and assume that CCis also “extreme”, i.e., C(u1t,…,udt)=(C(u))t∀t>0,∀u∈]0,1]d.C\left({u}_{1}^{t},\ldots ,{u}_{d}^{t})={\left(C\left(u))}^{t}\hspace{1em}\forall t\gt 0,\hspace{1em}\forall u\in {]0,1]}^{d}.Then, h(s)=(−logs)ph\left(s)={\left(-\log s)}^{p}for some p≥1p\ge 1.ProofWe only need to consider d=2d=2. It is easy to see that C(u)>0C\left(u)\gt 0for u∈]0,1]2u\in {]0,1]}^{2}. The corresponding STDF ℓ\ell is given as follows: ℓ(x,y)=−logC(e−x,e−y)=g−1(g(x)+g(y)),\ell \left(x,y)=-\log C\left({e}^{-x},{e}^{-y})={g}^{-1}\left(g\left(x)+g(y)),where g(x)≔h(e−x)g\left(x):= h\left({e}^{-x}). The preceding theorem implies g(x)=xpg\left(x)={x}^{p}for some p≥1p\ge 1; hence, h(s)=(−logs)ph\left(s)={\left(-\log s)}^{p}.□Remark 3In the definition of an Archimedean copula (as in Nelsen’s book [4]), it is not assumed that hhis unbounded; the case of a decreasing bijection h:]0,1]→[0,a[h:]0,1]\to {[}0,a{[}for some finite aais also allowed, with h−1{h}^{-1}extended to R+{{\mathbb{R}}}_{+}by h−1(x)≔0∀x≥a{h}^{-1}\left(x):= 0\hspace{1em}\forall x\ge a. But then a copula of the form C(u,v)=h−1(h(u)+h(v))C\left(u,v)={h}^{-1}\left(h\left(u)+h\left(v))cannot be extreme: choose u,v>0u,v\gt 0such that h(u)>a2h\left(u)\gt \frac{a}{2}, h(v)>a2h\left(v)\gt \frac{a}{2}, so that h(u)+h(v)>ah\left(u)+h\left(v)\gt a. Then ut→1{u}^{t}\to 1, vt→1{v}^{t}\to 1for t→0t\to 0, h(ut)→0h\left({u}^{t})\to 0, h(vt)→0h\left({v}^{t})\to 0and h−1(h(ut)+h(vt))→h−1(0)=1,t→0.{h}^{-1}(h\left({u}^{t})+h\left({v}^{t}))\to {h}^{-1}\left(0)=1,t\to 0.However, (h−1(h(u)+h(v)))t=0∀t>0.{({h}^{-1}\left(h\left(u)+h\left(v)))}^{t}=0\hspace{1.0em}\forall t\gt 0.Also so-called negative logistic (or Galambos) STDFs can be characterised by an “Archimedean property”, which however is not obvious at first sight. We remind that any STDF ℓ\ell on R+d{{\mathbb{R}}}_{+}^{d}is the co-survival function of some homogeneous Radon measure μ\mu on the locally compact space [0,∞]d⧹{∞d}≕Zd{\left[0,\infty ]}^{d}\setminus \left\{{\infty }_{d}\right\}\hspace{0.33em}=: \hspace{0.33em}{Z}_{d}, i.e., ℓ(x)=μ([x,∞]c)≕μˇ(x),x∈R+d.\ell \left(x)=\mu ({\left[x,\infty ]}^{c})\hspace{0.33em}=: \hspace{0.33em}\check{\mu }\left(x),\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d}.By definition of a Radon measure, μˇ(x)<∞∀x\check{\mu }\left(x)\lt \infty \hspace{1em}\forall x. The d.f. μ^(x)≔μ([0,x])\widehat{\mu }\left(x):= \mu \left(\left[0,x])of μ\mu is of course also finite and homogeneous.The family {fp∣p∈]−∞,0[}\{{f}_{p}| p\in ]-\infty ,0{[}\}of negative logistic STDFs is defined by fp(x)≔∑∅≠α⊆[d](−1)∣α∣+1∑i∈αxip1/p,x∈R+d.{f}_{p}\left(x):= \sum _{\varnothing \ne \alpha \subseteq \left[d]}{\left(-1)}^{| \alpha | +1}{\left(\sum _{i\in \alpha }{x}_{i}^{p}\right)}^{1\text{/}p},\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d}.Theorem 7Let ℓ\ell be a d-variate STDF, ℓ=μˇ\ell =\check{\mu }, with μ∈M+(Zd)\mu \in {M}_{+}\left({Z}_{d}), such that the d.f. μ^\widehat{\mu }is “Archimedean,” i.e., μ^(x)=g−1∑i=1dg(xi),x∈R+d,\widehat{\mu }\left(x)={g}^{-1}\left(\mathop{\sum }\limits_{i=1}^{d}g\left({x}_{i})\right),\hspace{1.0em}x\in {{\mathbb{R}}}_{+}^{d},where g:[0,∞]→[0,∞]g:\left[0,\infty ]\to \left[0,\infty ]is a continuous bijection. Then, ∃p<0\exists \hspace{-0.16em}p\lt 0such that g(x)=xpg\left(x)={x}^{p}, and ℓ=fp\ell ={f}_{p}.ProofBy assumption μ^(x1,…,xd−1,∞)=limxd→∞g−1∑i<dg(xi)+g(xd)\widehat{\mu }\left({x}_{1},\ldots ,{x}_{d-1},\infty )=\mathop{\mathrm{lim}}\limits_{{x}_{d}\to \infty }{g}^{-1}\left(\sum _{i\lt d}g\left({x}_{i})+g\left({x}_{d})\right)is finite; hence, g(∞)=0g\left(\infty )=0and ggis decreasing. By iteration, we obtain μ^(x1,x2,∞,…,∞)=g−1(g(x1)+g(x2)),\widehat{\mu }\left({x}_{1},{x}_{2},\infty ,\ldots ,\infty )={g}^{-1}\left(g\left({x}_{1})+g\left({x}_{2})),and from Theorem 5, we infer g(x)=xpg\left(x)={x}^{p}, where p<0p\lt 0, i.e., μ^(x)=∑i≤dxip1/p∀x∈R+d\widehat{\mu }\left(x)={\left({\sum }_{i\le d}{x}_{i}^{p}\right)}^{1\text{/}p}\hspace{1em}\forall x\in {{\mathbb{R}}}_{+}^{d}.The co-survival function μˇ\check{\mu }of μ\mu is easily expressed in terms of μ^\widehat{\mu }(where we use that boundaries of intervals are μ\mu -null sets, μ\mu being homogeneous, see [6], p. 248): with Bi≔{y∈Zd∣yi<xi}{B}_{i}:= \{y\in {Z}_{d}| {y}_{i}\lt {x}_{i}\}, we have μˇ(x)=μ([x,∞]c)=μ(⋃i≤dBi)=∑iμ(Bi)−∑i<jμ(Bi∩Bj)±⋯,\check{\mu }\left(x)=\mu \left({\left[x,\infty ]}^{c})=\mu \left(\bigcup _{i\le d}{B}_{i}\right)=\sum _{i}\mu \left({B}_{i})-\sum _{i\lt j}\mu \left({B}_{i}\cap {B}_{j})\pm \cdots \hspace{0.33em},and since μ(B1)=μ^(x1,∞,∞,…)\mu \left({B}_{1})=\widehat{\mu }\left({x}_{1},\infty ,\infty ,\ldots )etc., μ(B1∩B2)=μ^(x1,x2,∞,∞,…)\mu \left({B}_{1}\cap {B}_{2})=\widehat{\mu }\left({x}_{1},{x}_{2},\infty ,\infty ,\ldots )etc., we arrive at ℓ(x)=μˇ(x)=∑xi−∑i<j(xip+xjp)1/p±⋯=fp(x).□\hspace{10.2em}\ell \left(x)=\check{\mu }\left(x)=\sum {x}_{i}-\sum _{i\lt j}{({x}_{i}^{p}+{x}_{j}^{p})}^{1\text{/}p}\pm \cdots ={f}_{p}\left(x).\hspace{15em}\square Remark 4In the above theorem, we have μ^(x)=∑i=1dxip1/p≕μ^p(x).\widehat{\mu }\left(x)={\left(\mathop{\sum }\limits_{i=1}^{d}{x}_{i}^{p}\right)}^{1\text{/}p}\hspace{0.33em}=: \hspace{0.33em}{\widehat{\mu }}_{p}\left(x).In ref. [5], Theorem 6, it was shown that this function is a “bona fide” d.f. iff p∈−∞,1d−1∪1d−2,…,12,1p\in \left[-\infty ,\frac{1}{d-1}\right]\cup \left\{\frac{1}{d-2},\ldots ,\frac{1}{2},1\right\}. The question arises if for positive ppin this set, there is also a corresponding STDF: the answer is NO: μp{\mu }_{p}is then a Radon measure on R+d{{\mathbb{R}}}_{+}^{d}, not on Zd=[0,∞]d⧹{∞d}{Z}_{d}={\left[0,\infty ]}^{d}\setminus \left\{{\infty }_{d}\right\}, in fact μˇp(x)=∞{\check{\mu }}_{p}\left(x)=\infty for each x≠0x\ne 0in R+d{{\mathbb{R}}}_{+}^{d}. For p→0p\to 0, we obtain as limit μ^0(x)=(∏i=1dxi)1/d{\widehat{\mu }}_{0}\left(x)={({\prod }_{i=1}^{d}{x}_{i})}^{1\text{/}d}, and μ0{\mu }_{0}is likewise not a Radon measure on Zd{Z}_{d}.Remark 5Theorems 6 and 7 add to the many common features between Gumbel (logistic) and Galambos (negative logistic) STDFs resp. copulas, nicely described in ref. [2].We also want to characterise nested logistic STDFs, but here we are first confronted with the interesting general question of the “composebility” of several STDFs in its simplest (already non-trivial) form: if f,g,hf,g,hare bivariate STDFs, when is f∘(g×h)f\circ \left(g\times h)again a STDF? For logistic STDFs f=ℓr,g=ℓpf={\ell }_{r},g={\ell }_{p}and h=ℓqh={\ell }_{q}(r,p,q∈[1,∞[r,p,q\in {[}1,\infty {[}), a sufficient condition is well known: r≤pr\le pand r≤qr\le q. We shall show that this is necessary, too.Theorem 8Let r,p>0r,p\gt 0such thatℓ(x,y,z)≔xp+yppr+zrr\ell \left(x,y,z):= \sqrt[r]{{\sqrt[p]{{x}^{p}+{y}^{p}}}^{r}+{z}^{r}}is a STDF. Then, 1≤r≤p1\le r\le p.Proofr≥1r\ge 1and p≥1p\ge 1is clear. We start with 0≥D(1,1)(0,0)ℓ(⋅,⋅,t)=t−2(1+tr)1/r+(2r/p+tr)1/r0\ge {D}_{\left(1,1)}^{\left(0,0)}\ell \left(\cdot ,\cdot ,t)=t-2{\left(1+{t}^{r})}^{1\text{/}r}+{\left({2}^{r\text{/}p}+{t}^{r})}^{1\text{/}r}or (2r/p+tr)1/r+t≤2(1+tr)1/r{\left({2}^{r\text{/}p}+{t}^{r})}^{1\text{/}r}+t\le 2{\left(1+{t}^{r})}^{1\text{/}r}for any t≥0t\ge 0.Equivalently, 2r/p≤[2(1+tr)1/r−t]r−tr∀t>0.{2}^{r\text{/}p}\le {{[}2{\left(1+{t}^{r})}^{1\text{/}r}-t]}^{r}-{t}^{r}\hspace{1.0em}\forall t\gt 0.We now make use of the Binomial series (1+x)r=1+rx+r2x2+⋯{\left(1+x)}^{r}=1+rx+\left(\genfrac{}{}{0.0pt}{}{r}{2}\right){x}^{2}+\cdots valid for ∣x∣<1| x| \lt 1and all r∈Rr\in {\mathbb{R}}. We obtain [2(1+tr)1/r−t]r−tr=tr[2(1+t−r)1/r−1]r−tr=tr2(1+1r⋅t−r+1∕r2t−2r+⋯)−1r−tr=tr1+2r⋅t−r+21∕r2t−2r+⋯r−tr=tr1+2r⋅t−r(1+o(1))r−tr=tr1+2t−r(1+o(1))+r22(1+o(1))rtr2+⋯−tr=2(1+o(1))+O(1)/tr→2fort→∞.\begin{array}{rcl}{{[}2{\left(1+{t}^{r})}^{1\text{/}r}-t]}^{r}-{t}^{r}& =& {t}^{r}{{[}2{\left(1+{t}^{-r})}^{1\text{/}r}-1]}^{r}-{t}^{r}\\ & =& {t}^{r}{\left[2\left(1+\frac{1}{r}\cdot {t}^{-r}+\left(\genfrac{}{}{0.0pt}{}{1/r}{2}\right){t}^{-2r}+\cdots )-1\right]}^{r}-{t}^{r}\\ & =& {t}^{r}{\left[1+\frac{2}{r}\cdot {t}^{-r}+2\left(\genfrac{}{}{0.0pt}{}{1/r}{2}\right){t}^{-2r}+\cdots \right]}^{r}-{t}^{r}\\ & =& {t}^{r}{\left[1+\frac{2}{r}\cdot {t}^{-r}\left(1+{\mathcal{o}}\left(1))\right]}^{r}-{t}^{r}\\ & =& {t}^{r}\left[1+2{t}^{-r}\left(1+{\mathcal{o}}\left(1))+\left(\genfrac{}{}{0.0pt}{}{r}{2}\right){\left(\frac{2\left(1+{\mathcal{o}}\left(1))}{r{t}^{r}}\right)}^{2}+\cdots \hspace{0.33em}\right]-{t}^{r}\\ & =& 2\left(1+{\mathcal{o}}\left(1))+{\mathcal{O}}\left(1)\hspace{0.1em}\text{/}\hspace{0.1em}{t}^{r}\to 2\hspace{0.33em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}t\to \infty .\end{array}As a consequence, r/p≤1r\hspace{0.1em}\text{/}\hspace{0.1em}p\le 1, or r≤pr\le p.□We arrive at a complete characterisation for composite logistic STDFs.Theorem 9Let r,p1,…,pd∈[1,∞[r,{p}_{1},\ldots ,{p}_{d}\in {[}1,\infty {[}and α1∪⋯∪αd=[n]{\alpha }_{1}\cup \cdots \cup {\alpha }_{d}=\left[n]a partition, ∣αj∣≥2∀j| {\alpha }_{j}| \ge 2\hspace{1em}\forall j. Then ℓr∘(ℓp1×⋯×ℓpd){\ell }_{r}\circ \left({\ell }_{{p}_{1}}\times \cdots \times {\ell }_{{p}_{d}})is a STDF if and only if r≤pir\le {p}_{i}for all ii.ProofSufficiency is well known, see, e.g., ref. [8], p. 256. The other direction follows from the previous theorem by considering (for p1{p}_{1}) ℓr(ℓp1(xei+yej),ℓp2(z⋅ek),ℓp3(0),…),{\ell }_{r}\left({\ell }_{{p}_{1}}\left(x{e}_{i}+y{e}_{j}),{\ell }_{{p}_{2}}\left(z\cdot {e}_{k}),{\ell }_{{p}_{3}}\left(0),\ldots ),where i,j∈α1i,j\in {\alpha }_{1}and k∈α2k\in {\alpha }_{2}, which gives r≤p1r\le {p}_{1}.□The “nested” STDFs just considered allow the following “Archimedean” characterisation:Theorem 10Let f,g,hf,g,hbe continuous bijections of R+{{\mathbb{R}}}_{+}, m≥2m\ge 2, n≥2n\ge 2. Ifℓ(x,y)=f−1fg−1∑i≤mg(xi)+fh−1∑j≤nh(yj)\ell \left(x,y)={f}^{-1}\left\{f\left[{g}^{-1}\left(\sum _{i\le m}g\left({x}_{i})\right)\right]+f\left[{h}^{-1}\left(\sum _{j\le n}h({y}_{j})\right)\right]\right\}is a STDF on R+m×R+n{{\mathbb{R}}}_{+}^{m}\times {{\mathbb{R}}}_{+}^{n}, then ℓ=ℓr∘(ℓp×ℓq)\ell ={\ell }_{r}\circ \left({\ell }_{p}\times {\ell }_{q})for some r,p,q≥1r,p,q\ge 1, such that r≤pr\le pand r≤qr\le q(ℓr{\ell }_{r}bivariate).ProofPutting y=0y=0, we have ℓ(x,0)=g−1∑i≤mg(xi)\ell \left(x,0)={g}^{-1}\left({\sum }_{i\le m}g\left({x}_{i})\right); hence, g(s)=spg\left(s)={s}^{p}, where p≥1p\ge 1. Similarly, h(s)=sqh\left(s)={s}^{q}, with q≥1q\ge 1. For x=(x1,0,…,0)x=\left({x}_{1},0,\ldots ,0), y=(y1,0,…,0)y=({y}_{1},0,\ldots ,0), we obtain ℓ(x1,0,…,0,y1,0,…,0)=f−1(f(x1)+f(y1))\ell \left({x}_{1},0,\ldots ,0,{y}_{1},0,\ldots ,0)={f}^{-1}(f\left({x}_{1})+f({y}_{1})), and so f(s)=srf\left(s)={s}^{r}, where r≥1r\ge 1. By Theorem 9, finally, r≤pr\le pand r≤qr\le q.□Corollary 2For m,n≥2m,n\ge 2and decreasing bijections f,g,h:]0,1]→R+f,g,h:]0,1]\to {{\mathbb{R}}}_{+}, ifC(u,v)=f−1fg−1∑i≤mg(ui)+fh−1∑j≤nh(vj)C\left(u,v)={f}^{-1}\left\{f\left[{g}^{-1}\left(\sum _{i\le m}g\left({u}_{i})\right)\right]+f\left[{h}^{-1}\left(\sum _{j\le n}h\left({v}_{j})\right)\right]\right\}is an extreme value copula, thenC(u,v)=∑i(−logui)pr/p+∑j(−logvj)qr/q1/rC\left(u,v)={\left\{{\left[\sum _{i}{\left(-\log {u}_{i})}^{p}\right]}^{r\text{/}p}+{\left[\sum _{j}{\left(-\log {v}_{j})}^{q}\right]}^{r\text{/}q}\right\}}^{1\text{/}r}for some r,p,q≥1r,p,q\ge 1with r≤pr\le pand r≤qr\le q.This class of copulas of “composite Gumbel type” was already considered in ref. [7], p. 366, as particular examples of so-called generalised Archimedean copulas.

Journal

Dependence Modelingde Gruyter

Published: Jan 1, 2022

Keywords: multivariate extreme value distribution; stable tail dependence function, extremal coefficient; logistic; negative logistic; nested logistic; fully d -alternating; Archimedean property; 60E05; 62H05; 26B40

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