INTRODUCTION
Extreme Values Theory (EVT) is based on modelling and measuring events which
occur with very small probability. Mainly two methods have been developed in
this theory: the block maxima method (Beirlant et al.,
2005) and the PeacksOverThreshold (POT) approach (Coles,
2001; Resnick, 1987). The block maxima method is interested
to asymptotic behavior of the laws of the componentwise maxima appropriately
normalized under the condition that the univariate margins are independent and
identically distributed (iid). This method shows that these asymptotic laws
are the Multivariate Extreme Value Distributions (MEVDs). Suggested originally
by hydrologists, the POT approach is rather based on modelling of exceedances
of a random sample over a large threshold within a time period.
Earlier studies have developed statistical structures to describe the dependence
of the multivariate distributions arising from these two approaches. In many
latest books and reviews on the topic (Beirlant et al.,
2005; Gaume, 2005), it has been shown that no single
parametric family can summary the MEVDs like does do the Generalized Extreme
Value (GEV) family in the univariate EVT. Nevertheless, if the univariate margins
are given the dependence of these distributions can be characterized by equivalent
measures like Pickands dependence function, exponent measure or stable tail
dependence function (Degen, 2006). Furthermore, Tajvidi
has shown (Tajvidi, 1996) that for a sample of random
vectors the
law H of the exceedances over a large threshold is the multivariate Generalized
Pareto Distribution (GPD). Moreover, this excess distribution H is linked to
the asymptotic componentwise maxima model G of the same sample by Eq.
1:
The aim and particularity of this study is to build a new structure which describes the dependence of multivariate GPDs but under given conditions made on the marginal distributions. This new conditional dependence function enable us to characterize the basic parametric subfamilies of threedimensional GPDs.
MATERIALS AND METHODS
In this study, we consider the following problem: Let consider a situation
where X = {(X_{1},...,X_{n});n≥1} is a random vector with
a multivariate GPD function H and we are interested to model a structure which
describes both the dependence of H under the condition X_{I}>x_{0,I}
and the dependence of the survival function
of H under the condition X_{J}≤x_{0,J}; x_{0,I} and
x_{0,J} being given realizations of the complementary lower dimensional
margins X_{I} and X_{J} of X. Therefore, it is desirable to
model the structure which gives at any realization x = (x_{1},...,x_{n})
of X the probability of the discordances For
this purpose Pickands dependence function of a MEVD would be useful (Coles,
2001; Beirlant et al., 2005). Let:
be the componentwise maxima of a random vector with
univariate iid variables and with distribution function F. A mdimensional continuous
and nondegenerate function G is a MEVD if there exists vectors of normalizing
sequences with
such
that (in componentwise algebraic notations), for all
If Eq. 2 holds F is said to belong to the maxdomain of attraction of G. Therefore, from the link established between G and H by Eq. 1, we obtain, for all x = (x_{1},...,x_{m})εIR^{m}, the following characterization:
where, A is the Pickands dependence function of H defined on the unit simplex
in IR^{m1} such that:
verifying for tεS_{m1}, the condition:
μ being the angular measure of H on S_{m1} (Beirlant
et al., 2005). In addition, the y_{i} are defined by the
transformations:
where, μ_{i}εIR. ξ_{i}. εIR and σ_{i}>0 are respectively the location, shape and scale parameters of the univariate margins G_{i} of G. The different values of the parameter ξ_{i} allow the GEV distribution defined by:
to describe the three types of asymptotic extreme behavior such as:
The laws Λ, Φ and Ψ are from Gumbel, Fréchet and Weibull, respectively.
RESULTS
Here, the main three theorems of this study will be presented and proved.
Angular Distribution of Multivariate GPDs
The following theorem gives the angular distribution of a multivariate GPD.
Theorem 1
Let H be a multivariate GPD. Then, there exists a function L(.) defined
on S_{m1} in IR^{m1} such as, for all x = (x_{1},...,x_{m})εIR^{m}:
Moreover, if H is continuously differentiable of order m, the density function l of L fulfills, for all t = (t_{1},...,t_{m1})εS_{m1}, the equality:
The function L(.) is the angular distribution of the multivariate GPD H.
Proof
Let V be the exponent measure function of the distribution H with unit Fréchet
margins (Michel, 2006; Resnick, 1987).
Therefore, for all x_{i}>0 we have:
It is known that:
Where:
l(.) being the angular density of H. Furthermore, we have:
This result inserted in Eq. 11 gives :
By taking:
We have:
Replacing l(t_{1},...,t_{m1}) in Eq. 10 we see that Eq. 9 assertion holds.
For proving the following theorems we define a conditional dependence measure for the family of multivariate distributions.
A Conditional Measure of a Multivariate Distribution
Let n, k be natural numbers such that {n≥2;1≤k<n} and let N_{k}
be a given subset of k elements of N = {1,...,n}, the set of the first n natural
numbers.
Definition 1
We define N_{k}partition of a random vector X ={(X_{1},...,X_{n}),
n≥2} (or the partition of X in the direction of N_{k}) by the pairwise
vector
as:
• 
is the kdimensional marginal vector of X whose component indexes are ordered
in the subset N_{k} 
• 
is the (nk)dimensional marginal vector of X whose component indexes are
ordered in the
complementary of N_{k} in N 

Similarly, every realization x = (x_{1},..,x_{n}) of X
can be decomposed into two parts 
are, respectively realizations of vectors et
.
If H, HN_{k} and denote
the distribution functions of the random vectors
then for all realization x = (x_{1},...,x_{n}) of X we have
are the upper endpoints of the functions HN_{k} and
Definition 2
Given a N_{k}partition of
X = (X_{l},...,X_{n}) we define the upper N_{k}discordance
degree of X as the conditional probability given for all x = (x_{l},...,
x_{n}) εIR^{n} by Similarly,
the lower N_{k}discordance degree of X is defined, for all x = (x_{l},...,x_{n})
in IR^{n} by .
The following definition characterizes the probability that one of the margins
and
exceeds
1/2, while the values taken by the other are less than 1/2.
Definition 3
Given the distribution H of a multivariate random vector X = {(X_{1},...,X_{n}),
n≥2} with univariate margins H_{i}, 1≤i≤n we define the upper
N_{k}median discordance degree of H by the real number denoted by :
s
quantile function of H_{i}. Similarly, the lower N_{k}median
discordance degree of H is defined by
Example 1
Let X = (X_{1}, X_{2}, X_{3}) be a trivariate random
vector. Let’s consider N_{2} = {1, 3}. The lower N_{2}discordance
degree of X is given, for x = (x_{1}, x_{2}, x_{3})
ε IR^{3} by .
Particularly, if H is a continuous distribution function of X we verify easily
that ,
while for N_{1} = {1}, the upper N_{1}median discordance degree
is given by
where H_{2}>0 and H_{1,3}>0 are, respectively the distribution
functions of the margins X_{2} and (X_{1}, X_{3});
being the inverse of the survival function of H_{i}. Let’s suppose,
in addition that:
for all x_{i} ε]0,1] (distribution whose univariate margins are uniform on 0,1]), we check easily that, for θ = 1/2 we get :
The following result shows that the N_{k}marginal distribution of a MEVD is also a MEVD.
Theorem 2
Suppose there exists a MEVD G describing the asymptotic behavior of the
componentwise maxima of X suitably normalized. Then, there exists a kdimensional
MEVD G_{k} and a (nk)dimensional MEVD G_{nk} associated,
respectively to the componentwise maxima of the marginal vectors and
Moreover
G_{k} and G_{nk} are the marginal distributions of G.
Proof
Let σ_{n} = (σ_{1},...,σ_{n})εIR^{n};
σ_{i}>0 and μ_{n} = (μ_{1};...;μ_{n})
εIR^{n} be the vectors of normalizing sequences of the componentwise
maxima M_{n} = (M_{1},...,M_{n}) associated to the MEVD
G by previous Eq. 2. Then, if
is the upper endpoint of marginal vector we
have:
Therefore, there exists two vectors of normalizing sequences σi,
N_{k}>0 and μN_{k} = (μ1,N_{k};...;μk,
N_{k}) ε IR^{k} such as the marginal componentwise maxima
MN_{k} = (M1, N_{k},...,Mk,N_{k}) of M_{n} converges
to GN_{k} according to Eq. 2. Thereby G_{k}
is a MEVD with k variables.
Similarly, we establish that G_{nk} also arises as the limiting distribution
of the marginal componentwise maxima linearly
normalized with vectors of sequences
with
A Conditional Dependence Function of a Multivariate GPD
Note that, in the above, each of the discordance degrees and
of
a random vector X can be obtained by functional transformations of the other.
Therefore, the following characterizations will be restricted to the upper which
will be denoted by δ in the simplest case
i.e., k = 1.
Theorem 3
Let G be a MEVD with discordance degree δ. Then, there exists a convex
function D defined on the unit simplex S_{m1} by:
for all (x_{1},...,x_{m}) in IR^{n} ; where the y_{i}(x_{i}) satisfy Eq. 4, for i = 1,…, m.
D is called the discordance function of G or of its corresponding GPD H.
Taking a MEVD with unit Gumbel margin, G_{i}(x_{i}) = exp{exp(x_{i})};
x_{i}>0, the following corollary characterizes the simplest upper
median degree, .
Corollary
Let G be a MEVD with unit Gumbel. Then, the upper median discordande degree
of G, denoted by is
given by
Example 2
Let G_{θ}, θ>1 be the logistic model of MEVD given
for (x_{1}, x_{2}, x_{3})εIR^{3} by
y_{i}(x_{i}) satisfying Eq. 4. Its discordance function is given, for all (t_{1}, t_{2})εS_{2} by
and the median discordance degree
Particularly, for θ = 2, we get =
0.864.
Proof of Theorem 3
Let's suppose that, for i = 1,…,m, the univariate margins G_{i}
of the MEVD G have the generalized form: G_{i}(x_{i}) = exp{y_{i}(x_{i})}
where the y_{i} (x_{i}) satisfy Eq. 4. Therefore,
for all (x_{l},...,x_{n})εIR^{n};
Thus, being
the joint distribution of the margin vector (X_{2},...,X_{n})
of X, we have:
Furthermore, due to theorem 1, the function is
a kdimensional extreme value distribution. Therefore, if A and are
the Pickands dependence functions of G and respectively,
then, in Eq. 12 we have:
Furthermore, we have:
where, D is the convex function such that 0≤1t_{l}≤1 and defined on the unit simplex
by
for all tεS_{m1}. Particularly for the trivariate case, we have:
defined on S_{2}.
APPLICATION TO THE TRIVARIATE MODELS OF GPDs
The logistic model is the most important family of multivariate GPDs.
The Family of Trivariate GPD of Logistic Type
Let X = (X_{1}, X_{2}, X_{3}) be a trivariate random
vector with a parametric distribution H_{θ}, θ>1. The above
Eq. 3 enable us to characterize H_{θ} by its
discordance function D_{θ} via its Pickands dependence function
A_{θ} (Michel, 2006).
Definition 4
The trivariate parametric function H_{θ} is a MGPD of Logistic
Type if H_{θ} has, for all (x_{1}, x_{2}, x_{3})εIR^{3}
the representation:
With the y_{i}(x_{i}) satisfying Eq. 4 and where the discordance function D_{θ} of H_{θ} is given for all:
We give here three basic trivariate GPDs of Logistic Type (Joe,
1997; Husler and Reiss, 1989) and we build their discordance
functions:
• 
The trivariate family of GPD of Logistic Type of Gumbel 
Particularly if θ→1¯ we obtain the trivariate Pareto independent model H(x_{1}, x_{2}, x_{3})=1+{y_{1}(x_{1}) +y_{2}(x_{2})+y_{3}(x_{3})]} with D(t_{1}, t_{2}) = 2t_{1} for (t_{1}, t_{2})εS_{2}
• 
The trivariate family of GPD of Logistic Type of Galambos 
• 
The trivariate family of GPD of Logistic Type of HuslerRéiss 
The GPD which describes the behavior of the exceedances of the trivariate normal distribution over a large threshold is given for (x_{1}, x_{2}, x_{3})εIR^{3}; θ = (θ_{1}, θ_{2}, θ_{3}) by:
where, Φ notes the distribution function of the standard normal law and
the
survival function of the bivariate normal distribution function with covariance
matrix:
Thus, for all (t_{1}, t_{2}) the corresponding discordance function is:
where, R(t_{1}, t_{2}, θ) is an integral rest defined for all (t_{1}, t_{2})εS_{2}.
The Family of Trivariate GPD of Nested Type
The Nested Logistic Type is an asymmetric subfamily of logistic model. It
generalizes this model to allow different degrees of dependence between the
components of the underlying random vector. For (x_{1}, x_{2},
x_{3})εIR^{3} and θ_{1}, θ_{2}>1,
define now, recursively the following norm
where, ._{θ} is the usual θnorm with the convention
that the absolute value is taken if the norm does not have an index (Joe,
1997; Michel, 2006).
Definition 5
The distribution function given, for all (x_{1}, x_{2},
x_{3})εIR^{3}, by
is called the generalized Pareto distribution of Nested logistic type.
The basic trivariate GPD of Nested Logistic Type is given, for θ_{1}, θ_{2}≥1 by:
The Family of Trivariate GPD of Asymmetric Logistic Type
The asymmetric distributions arise as the models which describe the asymptotic
behavior of the maxima of storms recorded at different locations along a coastline
(Gaume, 2005). They generalize the logistic model but
does not include the nested logistic model from the previous section (Michel,
2006).
Definition 6
Let B be a nonempty subset of {1,2,3} and let λ_{C}≥1 be
arbitrary numbers for every C⊂B with C≥2 and λ_{C} = 1
if C = 1. Furthermore, let 0≤p_{i,C}≤1 where, p_{i,C}
= 0 if I∉C and the side condition
is fulfilled for i=1,2,3. Then the distribution function
is a generalized Pareto distribution of Asymmetric Logistic Type.
The basic trivariate GPDs of Asymmetric Logistic Type with their discordance
functions follow (Joe, 1997):
• 
The trivariate Asymmetric GPD of Logistic Type of Gumbel 
• 
The trivariate Asymmetric GPD of Logistic Type of Galambos 
for (x_{1}, x_{2}, x_{3})εIR^{3} and θ_{1}, θ_{2}>0. We have, for (t_{1}, t_{2}), εS_{2}
DISCUSSION
The results of the study show that the dependence of all trivariate GDP, under
given conditions on the lower dimensional margins, is totally described by its
discordance function. These results are similar to the characterizations of
the multivariate GPDs developed by Tajvidi (1996) or to
the equivalent dependence measures for MEVDs (Resnick, 1987;
Beirlant et al., 2005). But the particularity of
this study is the that the new measure and function describe the joint dependence
under any condition made on the support of a lower dimensional margin. Moreover,
the applications of the study determine clearly the three main subfamilies of
the models of trivariate GPDs by characterizing them by their discordance function.
We found that the results conform to the solution of the problem considered earlier. This is seen at all realisations x = (x_{l},...,x_{n}) of the random vector X. We also note that the theorem 3 establishes a link between the new dependence structure and the previous dependence measures via Pickands dependence one.
CONCLUSION
In this research we have investigated about characterization of a conditional dependence of multivariate families of generalized Pareto distributions. We have built a new measure and function which describe this conditional dependence. Basic trivariate subfamilies of multivariate GPDs have been characterized by this function. Moreover, we have computed the expressions of this function for the usual trivariate subfamilies of GPDs.