We prove some important properties of the extremal coefficients of a stable tail dependence function (“STDF”) and characterise logistic and some related STDFs. The well known sufficient conditions for composebility of logistic STDFs are shown to be also necessary.
A multivariate extreme value (MEV) distribution (in a standardised form) is given by a distribution function (“d.f.”) on with the decisive property
and with standard one-dimensional Fréchet margins, defined by the d.f. for . The d.f. is in a one-to-one correspondence with its associated stable tail dependence function (“STDF”), defined by
where , and these STDFs allow an intrinsic characterisation: is a STDF iff is homogeneous ( ), normalised ( unit vector ), and “fully -alternating” (to be explained later on), cf. , Theorem 6.
The marginals of are given by
for . is again a MEV distribution with STDF . If has the d.f. , the subvector has d.f. .
Two main subjects will be treated in this article. The first one is about the so-called extremal coefficients of a ( -variate) STDF , defined by
(slightly abusing notation). Although is plainly not determined by its restriction to , 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 .
, , , ,
, for , , are the usual unit vectors in ,
is the set of Radon measures on a locally compact space
d.f. = distribution function.
2 Fully d-alternating functions
To define this notion, which is of particular importance in this article, we introduce a special notation for multivariate real-valued functions. Let be non-empty sets, , and . First, for and , we put , , and so for
i.e., another element of , being for and for . Also, for . We then define
Note that . For and , we define a “partial version” of with fixed values in the variables by
(For , this would be , and for , the constant .)
There is a two-step procedure to determine which will be needed later on:
Let , , , and define by
Let be non-empty, . Then is fully d-alternating (in symbols “ ”) iff for (both in ), and if also for each and (both in ), and for all .
This property is specific for co-survival functions, i.e., , e.g., for uniform on
but it is of special importance also for some infinite measures, as we will see shortly.
There is a more general notion of -alternating (“ ”) functions, with , cf. , describing monotonicity conditions of higher orders, not needed in this work.
For , i.e., , a function is a STDF iff is homogeneous (i.e., ), and normalised, i.e., for each unit vector. Disregarding normalisation, we consider
This set, obviously compact and convex, was shown in ref.  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
where , and for each homogeneous function ( ), we have the unique integral representation
with a probability measure on .
The function is then the so-called co-survival function of a homogeneous Radon measure on the locally compact space , i.e.,
(which is finite by the definition of a Radon measure).
3 Properties depending on the extremal coefficients
Let be a -variate STDF. Its restriction gives the so-called extremal coefficients for (hence, ). From the integral representation,
holds for any -variate STDF , and therefore,
Let be a -variate STDF. Then,
If , then ; hence, , i.e., and .
If , then for each ; hence,
i.e., -a.s., or . From
we deduce , or .□
Let have the MEV-distribution associated with the STDF , i.e., with d.f. , for . For , the subvector then has the d.f.
including , .
Condition (i) in Theorem 1 means that a.s. , and (ii) is equivalent with being iid (standard Fréchet). The independence of two subvectors of also depends only on the extremal coefficients, as we now shall see.
For disjoint (non-empty) subsets , the following properties are equivalent:
and are independent.
In view of the connection between and , only (i) (ii) has to be shown. Without restriction , i.e., . So, let us assume (i), then from
Let , , . Then , , , . Since , we obtain , or , and . It follows
Before we extend Theorem 2 to more than two subvectors, we need the following.
Let be any non-degenerate interval, function, and a partition with non-empty . Define by . Then is . If is homogeneous, so is .
being trivial, assume . It is clearly enough to consider the partition
since the general case then follows easily by iteration.
We have . Let with , define , and , then by Lemma 1,
as the sum of two non-positive numbers.□
The following result is a considerable generalisation of Theorem 1 (ii) and Theorem 2.
Let be a -variate STDF, and let be disjoint non-empty subsets of . The random vector is supposed to have the d.f. , . Then the following conditions are equivalent:
are pairwise independent.
Without restriction, we assume .
(i) (ii): We use induction, the case being true by Theorem 2. Supposing the conclusion for , we use ’s subadditivity to obtain
and hence, , are independent, and may be applied to and .
(iii) (i): We use again induction. For , there is nothing to prove. We assume validity for some and consider the case . Let
and define on by . By Lemma 2, is (not normalised!). Therefore,
Considering in Theorem 3 the special case , , we are back to Theorem 1 (ii), with the additional equivalence to pairwise independence, i.e., . 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.
Let be a d-variate STDF; such that . If , then also .
Let be - (resp. -)variate STDFs, such that also is a STDF. Then , (and ).
Let be a -valued random vector with STDF . Then, if a.s. and , and because of , a.s. . That is, .
Again let have as its STDF. Then, a.s. , , i.e., a.s., leading to , and .□
For “overlapping variables”, this is different:
is a STDF, as is also (with )
With iid standard Fréchet random variables , , and , a stochastic model for these two STDFs would be the random vector , resp. .
Note, however, that
is not a STDF: .
4 Characterisation of logistic and related STDFs
Perhaps the best-known STDFs are the logistic ones, i.e., the family , defined by
Among all symmetric STDFs they are particular, depending on in an “additive way,” being a function of for some . We shall see that there are no other STDFs with this property besides the logistic ones.
We begin by solving a functional equation.
Let be homogeneous, , and let be a continuous bijection, such that and
Then, such that (which of course extends uniquely to ).
Obviously and , and is either (strictly) increasing or decreasing. Since is also continuous, so is .
For , we have and (i.e., , in accordance with ). The equality shows to belong to
a multiplicative subgroup of as is easily seen. Hence, .
and for ,
i.e., also , where , and this converges to for .
This implies to be dense in : it suffices to show
and this follows because for any ,
(choose with , then ). If is increasing, then for some (negative!) , and for decreasing , we may choose instead .
Now is closed, being continuous; hence, and . It is well known that this implies for some . (For , we have ; this is the standard Cauchy equation, and being continuous, it has the form with ; therefore, .) From , we obtain
(Characterisation of logistic STDFs) Let be a d-variate STDF of the form
for some continuous bijection , without restriction.
Then, for some , i.e., .
Obviously is (strictly) increasing, in particular , and it suffices to consider . By the preceding theorem, for some , and implies , and
This result, assuming from the outset (though tacitly) the function to 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. . We state this as a corollary, being slightly more general while not assuming differentiability:
Let be a -variate Archimedean copula, i.e.,
with a decreasing bijection , and assume that is also “extreme”, i.e.,
Then, for some .
We only need to consider . It is easy to see that for . The corresponding STDF is given as follows:
where . The preceding theorem implies for some ; hence, .□
In the definition of an Archimedean copula (as in Nelsen’s book ), it is not assumed that is unbounded; the case of a decreasing bijection
for some finite is also allowed, with extended to by . But then a copula of the form
cannot be extreme: choose such that , , so that . Then , for , , and
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 on is the co-survival function of some homogeneous Radon measure on the locally compact space , i.e.,
By definition of a Radon measure, . The d.f. of is of course also finite and homogeneous.
The family of negative logistic STDFs is defined by
Let be a d-variate STDF, , with , such that the d.f. is “Archimedean,” i.e.,
where is a continuous bijection. Then, such that , and .
is finite; hence, and is decreasing. By iteration, we obtain
and from Theorem 5, we infer , where , i.e., .
The co-survival function of is easily expressed in terms of (where we use that boundaries of intervals are -null sets, being homogeneous, see , p. 248): with , we have
and since etc., etc., we arrive at
In the above theorem, we have
In ref. , Theorem 6, it was shown that this function is a “bona fide” d.f. iff . The question arises if for positive in this set, there is also a corresponding STDF: the answer is NO: is then a Radon measure on , not on , in fact for each in . For , we obtain as limit , and is likewise not a Radon measure on .
Theorems 6 and 7 add to the many common features between Gumbel (logistic) and Galambos (negative logistic) STDFs resp. copulas, nicely described in ref. .
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 are bivariate STDFs, when is again a STDF? For logistic STDFs and ( ), a sufficient condition is well known: and . We shall show that this is necessary, too.
Let such that
is a STDF. Then, .
and is clear. We start with
for any .
We now make use of the Binomial series
valid for and all . We obtain
As a consequence, , or .□
We arrive at a complete characterisation for composite logistic STDFs.
Let and a partition, . Then is a STDF if and only if for all .
Sufficiency is well known, see, e.g., ref. , p. 256. The other direction follows from the previous theorem by considering (for )
where and , which gives .□
The “nested” STDFs just considered allow the following “Archimedean” characterisation:
Let be continuous bijections of , , . If
is a STDF on , then for some , such that and ( bivariate).
Putting , we have ; hence, , where . Similarly, , with . For , , we obtain , and so , where . By Theorem 9, finally, and .□
For and decreasing bijections , if
is an extreme value copula, then
for some with and .
This class of copulas of “composite Gumbel type” was already considered in ref. , p. 366, as particular examples of so-called generalised Archimedean copulas.
Conflict of interest: The author states no conflict of interest.
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