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Conditional Dependence of Trivariate Generalized Pareto Distributions



Diakarya Barro
 
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ABSTRACT

In this study we consider the dependence of the family of multivariate generalized Pareto distributions under given conditions on lower dimensional margins. A new function which describes this conditional dependence is built via Pickands dependence function. This function provides a new characterization of the basic subfamilies of trivariate generalized Pareto distributions.

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Diakarya Barro , 2009. Conditional Dependence of Trivariate Generalized Pareto Distributions. Asian Journal of Mathematics & Statistics, 2: 20-32.

DOI: 10.3923/ajms.2009.20.32

URL: https://scialert.net/abstract/?doi=ajms.2009.20.32
 

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 Peacks-Over-Threshold (POT) approach (Coles, 2001; Resnick, 1987). The block maxima method is interested to asymptotic behavior of the laws of the component-wise 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 Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsthe 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 component-wise maxima model G of the same sample by Eq. 1:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(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 three-dimensional GPDs.

MATERIALS AND METHODS

In this study, we consider the following problem: Let consider a situation where X = {(X1,...,Xn);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 XI>x0,I and the dependence of the survival function of H under the condition XJ≤x0,J; x0,I and x0,J being given realizations of the complementary lower dimensional margins XI and XJ of X. Therefore, it is desirable to model the structure which gives at any realization x = (x1,...,xn) of X the probability of the discordances Image for - Conditional Dependence of Trivariate Generalized Pareto DistributionsFor this purpose Pickands dependence function of a MEVD would be useful (Coles, 2001; Beirlant et al., 2005). Let:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

be the component-wise maxima of a random vector Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionswith univariate iid variables and with distribution function F. A m-dimensional continuous and non-degenerate function G is a MEVD if there exists vectors of normalizing sequences Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionswith Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionssuch that (in component-wise algebraic notations), for all Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(2)

If Eq. 2 holds F is said to belong to the max-domain of attraction of G. Therefore, from the link established between G and H by Eq. 1, we obtain, for all x = (x1,...,xm)εIRm, the following characterization:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(3)

where, A is the Pickands dependence function of H defined on the unit simplex

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

in IRm-1 such that:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

verifying for tεSm-1, the condition:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

μ being the angular measure of H on Sm-1 (Beirlant et al., 2005). In addition, the yi are defined by the transformations:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(4)

where, μiεIR. ξi. εIR and σi>0 are respectively the location, shape and scale parameters of the univariate margins Gi of G. The different values of the parameter ξi allow the GEV distribution defined by:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(5)

to describe the three types of asymptotic extreme behavior such as:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(6)

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 Sm-1 in IRm-1 such as, for all x = (x1,...,xm)εIRm:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(8)

Moreover, if H is continuously differentiable of order m, the density function l of L fulfills, for all t = (t1,...,tm-1)εSm-1, the equality:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(9)

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 xi>0 we have:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(10)

It is known that:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(11)

Where:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

l(.) being the angular density of H. Furthermore, we have:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

This result inserted in Eq. 11 gives :

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

By taking:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

We have:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Replacing l(t1,...,tm-1) 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 Nk be a given subset of k elements of N = {1,...,n}, the set of the first n natural numbers.

Definition 1
We define Nk-partition of a random vector X ={(X1,...,Xn), n≥2} (or the partition of X in the direction of Nk) by the pairwise vectorImage for - Conditional Dependence of Trivariate Generalized Pareto Distributions as:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
is the k-dimensional marginal vector of X whose component indexes are ordered in the subset Nk
Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
is the (n-k)-dimensional marginal vector of X whose component indexes are ordered in Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsthe complementary of Nk in N
  Similarly, every realization x = (x1,..,xn) of X can be decomposed into two parts

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

are, respectively realizations of vectors Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionset Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions. If H, HNk and Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsdenote the distribution functions of the random vectors Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions then for all realization x = (x1,...,xn) of X we have

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

are the upper endpoints of the functions HNk and Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Definition 2
Given a Nk-partition Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsof X = (Xl,...,Xn) we define the upper Nk-discordance degree of X as the conditional probability given for all x = (xl,..., xn) εIRn by Image for - Conditional Dependence of Trivariate Generalized Pareto DistributionsSimilarly, the lower Nk-discordance degree of X is defined, for all x = (xl,...,xn) in IRn by Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions.

The following definition characterizes the probability that one of the margins Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsand Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsexceeds 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 = {(X1,...,Xn), n≥2} with univariate margins Hi, 1≤i≤n we define the upper Nk-median discordance degree of H by the real number denoted by : Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionss quantile function of Hi. Similarly, the lower Nk-median discordance degree of H is defined by Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Example 1
Let X = (X1, X2, X3) be a trivariate random vector. Let’s consider N2 = {1, 3}. The lower N2-discordance degree of X is given, for x = (x1, x2, x3) ε IR3 by Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions. Particularly, if H is a continuous distribution function of X we verify easily that Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions, while for N1 = {1}, the upper N1-median discordance degree is given by Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions where H2>0 and H1,3>0 are, respectively the distribution functions of the margins X2 and (X1, X3); Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions being the inverse of the survival function of Hi. Let’s suppose, in addition that:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

for all xi ε]0,1] (distribution whose univariate margins are uniform on 0,1]), we check easily that, for θ = 1/2 we get :

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

The following result shows that the Nk-marginal distribution of a MEVD is also a MEVD.

Theorem 2
Suppose there exists a MEVD G describing the asymptotic behavior of the component-wise maxima of X suitably normalized. Then, there exists a k-dimensional MEVD Gk and a (n-k)-dimensional MEVD Gn-k associated, respectively to the component-wise maxima of the marginal vectors Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsand Image for - Conditional Dependence of Trivariate Generalized Pareto DistributionsMoreover Gk and Gn-k are the marginal distributions of G.

Proof
Let σn = (σ1,...,σn)εIRn; σi>0 and μn = (μ1;...;μn) εIRn be the vectors of normalizing sequences of the component-wise maxima Mn = (M1,...,Mn) associated to the MEVD G by previous Eq. 2. Then, if

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

is the upper endpoint of marginal vector Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionswe have:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Therefore, there exists two vectors of normalizing sequences Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsσi, Nk>0 and μNk = (μ1,Nk;...;μk, Nk) ε IRk such as the marginal component-wise maxima MNk = (M1, Nk,...,Mk,Nk) of Mn converges to GNk according to Eq. 2. Thereby Gk is a MEVD with k variables.

Similarly, we establish that Gn-k also arises as the limiting distribution of the marginal component-wise maxima Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionslinearly normalized with vectors of sequences Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions with Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

A Conditional Dependence Function of a Multivariate GPD
Note that, in the above, each of the discordance degrees Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsand Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsof a random vector X can be obtained by functional transformations of the other. Therefore, the following characterizations will be restricted to the upper Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionswhich will be denoted by δ in the simplest caseImage for - Conditional Dependence of Trivariate Generalized Pareto Distributions 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 Sm-1 by:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(7)

for all (x1,...,xm) in IRn ; where the yi(xi) 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, Gi(xi) = exp{-exp(-xi)}; xi>0, the following corollary characterizes the simplest upper median degree, Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions.

Corollary
Let G be a MEVD with unit Gumbel. Then, the upper median discordande degree of G, denoted by Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsis given by

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Example 2
Let Gθ, θ>1 be the logistic model of MEVD given for (x1, x2, x3)εIR3 by

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

yi(xi) satisfying Eq. 4. Its discordance function is given, for all (t1, t2)εS2 by

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

and the median discordance degree

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Particularly, for θ = 2, we get Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions= 0.864.

Proof of Theorem 3
Let's suppose that, for i = 1,…,m, the univariate margins Gi of the MEVD G have the generalized form: Gi(xi) = exp{-yi(xi)} where the yi (xi) satisfy Eq. 4. Therefore, for all (xl,...,xn)εIRn;

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
(12)

Thus, Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsbeing the joint distribution of the margin vector (X2,...,Xn) of X, we have:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Furthermore, due to theorem 1, the function Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsis a k-dimensional extreme value distribution. Therefore, if A and Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsare the Pickands dependence functions of G and Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsrespectively, then, in Eq. 12 we have:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Furthermore, we have:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

where, D is the convex function such that 0≤1-tl≤1 and defined on the unit simplex

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

by

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

for all tεSm-1. Particularly for the trivariate case, we have:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

defined on S2.

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 = (X1, X2, X3) be a trivariate random vector with a parametric distribution Hθ, θ>1. The above Eq. 3 enable us to characterize Hθ by its discordance function Dθ via its Pickands dependence function Aθ (Michel, 2006).

Definition 4
The trivariate parametric function Hθ is a MGPD of Logistic Type if Hθ has, for all (x1, x2, x3)εIR3 the representation:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

With the yi(xi) satisfying Eq. 4 and where the discordance function Dθ of Hθ is given for all:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

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
Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Particularly if θ→1¯ we obtain the trivariate Pareto independent model H(x1, x2, x3)=1+{-y1(x1) +y2(x2)+y3(x3)]} with D(t1, t2) = 2-t1 for (t1, t2)εS2

The trivariate family of GPD of Logistic Type of Galambos
Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
The trivariate family of GPD of Logistic Type of Husler-Réiss

The GPD which describes the behavior of the exceedances of the trivariate normal distribution over a large threshold is given for (x1, x2, x3)εIR3; θ = (θ1, θ2, θ3) by:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

where, Φ notes the distribution function of the standard normal law and Image for - Conditional Dependence of Trivariate Generalized Pareto Distributionsthe survival function of the bivariate normal distribution function with covariance matrix:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

Thus, for all (t1, t2) the corresponding discordance function is:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

where, R(t1, t2, θ) is an integral rest defined for all (t1, t2)εS2.

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 (x1, x2, x3)εIR3 and θ1, θ2>1, define now, recursively the following norm

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

where, ||.||θ is the usual θ-norm 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 (x1, x2, x3)εIR3, by

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

is called the generalized Pareto distribution of Nested logistic type.

The basic trivariate GPD of Nested Logistic Type is given, for θ1, θ2≥1 by:

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

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 non-empty subset of {1,2,3} and let λC≥1 be arbitrary numbers for every C⊂B with |C|≥2 and λC = 1 if |C| = 1. Furthermore, let 0≤pi,C≤1 where, pi,C = 0 if I∉C and the side condition

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

is fulfilled for i=1,2,3. Then the distribution function

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

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
Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions
The trivariate Asymmetric GPD of Logistic Type of Galambos
Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

for (x1, x2, x3)εIR3 and θ1, θ2>0. We have, for (t1, t2), εS2

Image for - Conditional Dependence of Trivariate Generalized Pareto Distributions

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 = (xl,...,xn) 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.

REFERENCES

1:  Beirlant, J., Y. Goegebeur, J. Segers and J. Teugels, 2005. Statistics of Extremes Theory and Applications. John Wiley and Sons, Chichester, ISBN-13: 978-0-471-97647-9

2:  Coles, S., 2001. An Introduction to Statistical Modeling of Extreme Values. 1st Edn., Springer, New York, ISBN-13: 978-1852334598
Direct Link  |  

3:  De Haan, L. and A. Ferreira, 2006. Extreme Value Theory: An Introduction. Springer, Berlin

4:  Gaume, E., 2005. On the asymptotic behavior of flood peak distributions-theoretical Hydrol. Earth Syst. Sci. Discuss., 2: 1835-1864.
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5:  Degen, M., 2006. On multivariate generalised pareto distributions and high risk scenarios. Diploma Thesis, Department of Mathematics, ETH Zurich.

6:  Husler, J. and R.D. Reiss, 1989. Extreme Value Theory: Proceedings of a Conference Held in Oberwolfach Dec. 6-12, 1987. Springer, Berlin, ISBN-13: 9780387969541, pp: 279

7:  Joe, H., 1997. Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probabilty. Vol. 73, Chapman and Hall, London, ISBN-13: 9780412073311

8:  Michel, R., 2006. Simulation and Estimation in Multivariate Generalized Pareto. Dissertation, Fakultat fur Mathematik und Informatikn, Universitat Wurzburg, Wurzburg

9:  Resnick, S., 1987. Extreme Values, Regular Variation and Point Processes.Extreme Values, Regular Variation and Point Processes.Extreme Values, Regular Variation and Point Processes. Springer Series of the Applied Probability Trust, Springer, New York

10:  Tajvidi, N., 1996. Characterisation and Some Statistical Aspects of Univariate and Multivariate Generalised Pareto Distributions. Dissertation, Department of Mathematics, Chalmers Tekniska Hogskola Goteborg, Swedish

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