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UtilitasMathematica
ISSN 0315-3681 Volume 120, 2023
1184
Local and Interval Dependence for Bivariate Gamma distribution Created by
Archimedean Copula
Swar O. Ahmed1,2, Khwazbeen S. Fatah1 and Namam J. Mahmoud1
1 Mathematics Departement, Collage of Science, Salahaddin University-Erbil, Kurdistan Region,
Iraq.
2Corresponding Author:
E-mail: Swar.ahmed@su.edu.krd. Mobile: +964(0)7504972213
ABSTRACT
The Interval Dependence Structure (IDS) and Local Dependence Function (LDF) are tools used to
measure possible changes that occur in the correlation of bivariate random variables at each point
acrossthe whole domain. Bivariate distribution functionswith Gamma marginals are used to
construct correlated models that joint a widespread variety of data having positive values. In this
paper, to deriveformulas to measure the LDF and IDS, bivariate Gamma distribution are
constructedfor two different copula models (Frank, Clyton). Additionally, the key assumptions and
properties for each model are explained and then the two approaches are compared based on the
observed differences between the two measures, IDS and, LDF using an example from each model.
Key words: LDF, IDS, Gamma distribution, Bivariate distribution, Copula.
1- Introduction
The univariate gamma distribution, which has been studied extensively in scientific literature, is a
widely used statistical model to analyse skewed data in various fields of research; the exponential,
chi-square and Erlang probability distributions are special cases of the gamma distribution in
whichnumeroussimplifications and different forms of gamma density have been developed and
applied in different areas (Kotz, et al., 2004). The univariate gamma has been extended to the
bivariate case in many ways and several forms of bivariate gamma function have been obtained
(Izawa, 1965). In scientific literature, to construct bivariate distributions, several methods have been
proposed, see (Balakrishnan and Lai, 2009;Mathaland Moschopoulos, 1992;Joe et al., 2010), the
Bivariate gamma has found to be applicable in different areas. In this paper, to compute the local
dependence for bivariate gamma distribution, which is derived from copula function with marginals
obtained from gamma distributions, two measuresIDS and LDF with their formulas are studied.
Moreover, the copula functions presented in this paper are selected based on their common usage in
the multivariate research study, in addition to the ease of construction of their IDS and LDF as well
as their ability to capture a wide range of dependency structures. The main focus is on two models
from the Archimedean family (Nelsen,2007; Ahmed, et al., 2021).
2- Methodology
Copulas are tools used to join univariate functions(df) to construct multivariatedistribution
functions (Nelsen, 2007)Let and represent the marginals of Gamma distribution for
RV's and , respectively. Sklar's states: if F is a bivariate distribution function with marginals
and , then thereis a copula function
C:, where for all RV's and in ∞∞, there is:
The parameter , in the copula model,defines the degree ofcorrelation between the random
variables;it is defined by Kendall's correlation coefficient (Lee, et al., 2013; Hua, and Joe, 2011;
Koutoumanou, et al., 2017), and as follows:
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Another measure of correlationdefined as Pearson's correlation, which is denoted by, often
is not appropriate to sufficientlydefine the dependence structure of bivariate rv's (, ) with joint
distribution function. A remarkable review paper on bivariate and dependence is (Jones, 1996).
The Local Dependence measure, whichthoroughly explains the nature of joint variations,
investigates the main changes in the strength of the association along the range of (, ). To
demonstrate the notion of local dependence, there are several models and graphical representations;
for example local dependence maps (Jonesand Koch,2003);local Gaussian correlation
(TjøstheimandHufthammer,2013);Kendall plots (Genest and Boies, 2003) and chi-plots (Fisher and
Switzer, 1985). In this paper, the focus is on the definition of LDF as given by (Holland and Wang,
1987) and IDS given by (Esa andDimitrov, 2016); they are explained below.
The measure LD is defined as follows:
wherecomputes the amount of variation of the logarithm of the bivariate density at every
point of and . Thismeasure can be seen as Pearson correlation coefficientat each point (,
) (Jonesand Koch, 2003);it estimatesthe grade of the association between random variables and in
a neighbourhood of any point in the domain of f. For independent randomvariables,the LDF
always tends to zero, while it is constant for some distributions shown by (Jones, 1996), for instance
the LDF of the bivariate normal distribution
.
Besides, (Esa and Dimitrov, 2016) definedanother measure of local dependence called
Interval Dependence Structure (IDS); itdetermines the local dependence on the whole surface of
bivariate distribution function. IDS is definedby the following formula:
Where
Similarly, for
And
3- The local dependence for bivariate Gamma distribution
In this sectiontwo measure, IDS and LDF, for the bivariate distribution, which is constructed by
copula with Gamma marginals, are derived. Furthermore, copula probability distribution, the IDS
and LDF for each bivariate with varying parameters are illustrated; the software MATLAB 20b is
used for computations.
The general formula for the probability density function of the gamma distribution is
Γ
Where Γ
∞
Assume thatare random variables each with a Gamma distribution for their shape and scale
parameters , respectively, i.e.:
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γ
Γ
Where γ
The following, for each copula model, formulas and graphs (surface and contours) of the copula
function (cdf), copula density function (pdf), LDF and IDS for different marginal and dependence
parameters are shown. This illustrates how the same parameters can clearly lead to different
bivariate associations for differing copula functions. The surface and contours graph are shown a
bivariate model of (df, pdf, LDF and IDS) in a different perspective. For all graphs, the X and Y
axes represent the two RV's, respectively. The construction of the two copula models is
described below.
3.1- Frank Copula
Frank copula model is built through the subsequent mixture of marginal distributions
(Nelsen,2007),
where ∞∞ if then are independent.
The probability distribution function of the copula with Gamma marginals can be written as
follows:
The following figures show the bivariate function and probability function for Frank Copula for
different values of the parameter.
Figure 1.df and pdf of Frank copula for shape, Scale and dependence parameters:
.
And the measure LDF is defined as:
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To measure the interval dependence between the pair of random variables X1 and X2 on the
rectangle D= []×[] for , the IDSis determined as
follow:
Figure 2 shows the IDS, LDF as surface and contour maps of a bivariate Gammd distribution
formed through Frank model with given parameters.
Figure 2.IDS and LDF of bivariate Gammd distribution with parameters:
.
Figure 2, shows that LDF and the IDS display different patterns of dependence;the LDF shows that
the dependence is higher in the core of the random variables, while the pattern of the dependence
obtained by IDS divides the domain of random variables into four sectors in which the first quarter
gives higher value of dependence. Moreover, the shape of the measure LDF is like the shape of the
pdf mostly for bivariate Gamma distribution created by frank copula model.
3.2- Clayton Copula
The Clayton model is defined as follows(Nelsen,2007), where ∞
It is considered that 0<<, hence the model can be simplified to
Then, the probability density function of the copula model when the marginals are Gamma
distributions is defined as follow:
The following figure, Figure 3, presents the bivariate df and pdf of Clayton model with given
parameters.
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Figure 3.df and pdf of Clayton model with shape, scale and dependence parameters:
.
Figure 3illustrates that the pdf and cdf, of bivariate Gamma distribution via Clayton copula model,
have the same shape.
Furthermore, the LDF of Bivariate Gamma distribution created via Clayton Model are define as
follow:
To measure the interval dependence between a pair of random variables X1 and X2 on the rectangle
D= []×[], for, the IDSisdefined as follow:
In the following figure, Figure 4, the IDS, LDF surface and contour maps of a Clayton model with
specified parameters are presented.
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Figure 4.IDS and LDF of Clayton model with parameters:
.
From figure 4, it is noted that the LDF, of bivariate Gamma distribution formed by Clayton model,
has a strong dependence at the end tail of the distribution and the maximum value in the centre of
fourth quarter, while the IDS has shown a very distinct pattern of dependence.
4- Conclusions
In this paper, two measures of dependences known as LDF and IDS are defined, and their formulas
are derived for two Bivariate Gamma distribution, which is constructed by copula models (Frank,
Clayton); each can easily be programmed to produce IDS and LDF maps to explore and compare
dependence for bivariate Gamma distribution. The result shown that each measure displays a
different pattern of dependence for the same bivariate distribution, for Frank Models the LDF
shows that the dependence is higher only in the central of the random variables while the IDS
divides the domain of random variables into four sectors in which the first quarter gives higher
value of dependence. Furthermore, for Clayton copula, the LDF shows strong dependence at the
end tail of the distribution with the maximum value in the centre of fourth quarter, but the IDS
showsa distinct pattern of dependence.
References
Lai, C.D. and Balakrishnan, N., 2009. Continuous bivariate distributions. Springer-Verlag New
York. Kotz, S., Balakrishnan, N. and Johnson, N.L., 2004. Continuous multivariate distributions,
Volume 1: Models and applications (Vol. 1). John Wiley & Sons.
Izawa, T., 1965. Two or multi-dimensional gamma-type distribution and its application to rainfall
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Nadarajah, S. and Gupta, A.K., 2006. Some bivariate gamma distributions. Applied mathematics
letters, 19(8), pp.767-774.
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Mathal, A.M. and Moschopoulos, P.G., 1992. A form of multivariate gamma distribution. Annals of
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