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A note on canonical copulas
Carles Cuadras
To cite this version:
Carles Cuadras. A note on canonical copulas. 2017. �hal-01598825v3�
A note on canonical copulas
Carles M. Cuadras
University of Barcelona (Spain) November 15, 2017
Abstract
Cuadras (J. Mult. An.139, 28-44, 2015) proposed a method to con- struct parametric copulas, based on an integral representation, which uses canonical correlation functions. The generating function is im- proved and it is shown that some copulas can be constructed more generally.
AMS Subject Classi…cation: 62H20 (primary) 60E05 (secondary)
1 Introduction
A copula function C(u; v) is a bivariate cdf with uniform (0;1) marginals.
The copulas capture the dependence properties of two r.v.’s de…ned on the same probability space and have many applications.
In Cuadras (2015) a procedure was proposed for generating parametric copulas, as a consequence of the integral representation
C (u; v) = uv+ Z 1
maxfu;vg
f ( )(u= )(v= )d ; (1)
where f ; 2 I = [0;1]; is a continuous canonical function. The adjec- tive “canonical” is due to the interpretation of f as a canonical correlation function in a continuous diagonal expansion of a bivariate distribution.
For example, for the Cuadras-Augé family of copulas CA (u; v) = minfu; vg (uv)1 ; 2I;
the canonical correlations are described by a continuous function rather than a discrete sequence. Thus (1) holds forf ( ) = 1 :
The family of canonical copula functions (Cuadras, 2015), is given by CF (u; v) = uvG [Q(u; v)]=Q(u; v): (2) Here G ( ) = F ( ) + F (1); where F ( ) is a primitive of f ( ) and Q(u; v) =C1(u; v)=C2(u; v) is the quotient of two known copulas.
In this note we propose a simpler way for generating canonical copulas and show how to construct more copulas.
2 Generating function
Instead of the above functions F and G , we de…ne H ( ) =
Z 1f (t)
t2 dt+ 1:
Clearly H (1) = 1, H ( ) is decreasing in the interval (0;1] and f ( ) = 2H0( ):
The family of canonical copulas is now de…ned by
CF (u; v) =uvH [Q(u; v)]: (3)
Of course, (2) and (3) are equivalent.
3 Some examples
Many copulas can be generated combiningH and Q:
1. Cuadras-Augé family is generated by H ( ) = and the quotient Q(u; v) = (uv)=minfu; vg= maxfu; vg:
2. Durante family is de…ned by
D (u; v) = minfu; vgf(maxfu; vg):
This family was suggested in Cuadras and Augé (1981) and studied by Durante (2007) and Mazoet al. (2014). D is a particular case ofCF if we take H( ) =f( )= and Q(u; v) = maxfu; vg:
3. H ( ) = exp[ (1 )], Q(u; v) = lnulnv+ 1give the Gumbel-Barnett family (see Nelsen, 2006):
GB (u; v) = uvexp[ lnulnv]; 2I:
Note that Q(u; v) is not the quotient of two copulas.
4. H ( ) = exp[ (1 )],Q(u; v) = 1 (1 u)(1 v)give the Celebioglu- Cuadras family
CC (u; v) =uvexp[ (1 u)(1 v)]:
See Celebioglu (1997), Cuadras (2009).
5. H ( ) = exp[ (1 )], Q(u; v) = maxfu; vg give the (possibly) new family of copulas
CN(1)(u; v) = uvexp[ (1 maxfu:vg)]:
6. H ( ) = 1 + (2 = ) cos( =2); Q(u; v) = 1 (1 u)(1 v) give CN(2)(u; v) = uv[1 + (2 = ) cos[f1 (1 u)(1 v)g =2]:
7. H ( ) = 1 + (2 = ) cos( =2); Q(u; v) = maxfu; vg provide CN(3)(u; v) = uv[1 + (2 = ) cos(maxfu; vg =2)]:
8. H ( ) = expfsin[ (1 )]g; Q(u; v) = 1 (1 u)(1 v) give CN(4)(u; v) =uvexpfsin[ (1 u)(1 v)]g:
It is obvious that any copula C can be generated with H( ) = 1= and Q(u; v) =uv=C(u; v);but this is a self-generation. We must take two known copulas to generate a new one.
Each H comes from a canonical correlation function f . For instance, f ( ) = 2sin( =2)gives H ( ) = 1 + (2 = ) cos( =2):
Finally, it is worth noting that Q in (3) could not be the quotient of two copulas, It is an open question to determine the necessary and su¢ cient conditions for Q to construct a copula following this method.
4 Acknowledgements
Work supported in part by grant MTM2015-65016-C2-2-R (MINECO/FEDER)
5 References
1. Celebioglu, S. (1997) A way for generating comprehensive copulas.
Journal of the Institute of Science and Technology of Gazi University, 10, 57-61.
2. Cuadras, C. M. (2009) Constructing copula functions with weighted geometric means. Journal of Statistical Planning and Inference, 139, 3766-3772.
3. Cuadras, C. M. (2015) Contributions to the diagonal expansion of a bivariate copula with continuous extensions. Journal of Multivariate Analysis, 139, 28-44.
4. Cuadras, C. M., Augé, J. (1981) A continuous general multivariate distribution and its properties. Communications in Statistics-Theory and Methods, A10,339-353.
5. Durante, F. (2007) A new family of symmetric bivariate copulas. C.
R. Acad. Sci. Paris Ser. I, 344, 195-198.
6. Mazo, G., Girard, S., Forbes, F. (2016) A ‡exible and tractable class of one-factor copulas. Statistics and Computing, 26 (5), 965-979.
7. Nelsen, R. B. (2006) An Introduction to Copulas. 2nd Ed. Springer, New York.
