Quasi-concave copulas, asymmetry and transformations

Elisabetta Alvoni; Pier Luigi Papini

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 2, page 311-319
  • ISSN: 0010-2628

Abstract

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In this paper we consider a class of copulas, called quasi-concave; we compare them with other classes of copulas and we study conditions implying symmetry for them. Recently, a measure of asymmetry for copulas has been introduced and the maximum degree of asymmetry for them in this sense has been computed: see Nelsen R.B., Extremes of nonexchangeability, Statist. Papers 48 (2007), 329–336; Klement E.P., Mesiar R., How non-symmetric can a copula be?, Comment. Math. Univ. Carolin. 47 (2006), 141–148. Here we compute the maximum degree of asymmetry that quasi-concave copulas can have; we prove that the supremum of { | C ( x , y ) - C ( y , x ) | ; x , y in [ 0 , 1 ] ; C is quasi-concave} is 1 5 . Also, we show by suitable examples that such supremum is a maximum and we indicate copulas for which the maximum is achieved. Moreover, we show that the class of quasi-concave copulas is preserved by simple transformations, often considered in the literature.

How to cite

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Alvoni, Elisabetta, and Papini, Pier Luigi. "Quasi-concave copulas, asymmetry and transformations." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 311-319. <http://eudml.org/doc/250212>.

@article{Alvoni2007,
abstract = {In this paper we consider a class of copulas, called quasi-concave; we compare them with other classes of copulas and we study conditions implying symmetry for them. Recently, a measure of asymmetry for copulas has been introduced and the maximum degree of asymmetry for them in this sense has been computed: see Nelsen R.B., Extremes of nonexchangeability, Statist. Papers 48 (2007), 329–336; Klement E.P., Mesiar R., How non-symmetric can a copula be?, Comment. Math. Univ. Carolin. 47 (2006), 141–148. Here we compute the maximum degree of asymmetry that quasi-concave copulas can have; we prove that the supremum of $\lbrace |C(x,y)-C(y,x)|; x,y$ in $[0,1]$; $C$ is quasi-concave\} is $\frac\{1\}\{5\}$. Also, we show by suitable examples that such supremum is a maximum and we indicate copulas for which the maximum is achieved. Moreover, we show that the class of quasi-concave copulas is preserved by simple transformations, often considered in the literature.},
author = {Alvoni, Elisabetta, Papini, Pier Luigi},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {copula; quasi-concave; asymmetry},
language = {eng},
number = {2},
pages = {311-319},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Quasi-concave copulas, asymmetry and transformations},
url = {http://eudml.org/doc/250212},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Alvoni, Elisabetta
AU - Papini, Pier Luigi
TI - Quasi-concave copulas, asymmetry and transformations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 2
SP - 311
EP - 319
AB - In this paper we consider a class of copulas, called quasi-concave; we compare them with other classes of copulas and we study conditions implying symmetry for them. Recently, a measure of asymmetry for copulas has been introduced and the maximum degree of asymmetry for them in this sense has been computed: see Nelsen R.B., Extremes of nonexchangeability, Statist. Papers 48 (2007), 329–336; Klement E.P., Mesiar R., How non-symmetric can a copula be?, Comment. Math. Univ. Carolin. 47 (2006), 141–148. Here we compute the maximum degree of asymmetry that quasi-concave copulas can have; we prove that the supremum of $\lbrace |C(x,y)-C(y,x)|; x,y$ in $[0,1]$; $C$ is quasi-concave} is $\frac{1}{5}$. Also, we show by suitable examples that such supremum is a maximum and we indicate copulas for which the maximum is achieved. Moreover, we show that the class of quasi-concave copulas is preserved by simple transformations, often considered in the literature.
LA - eng
KW - copula; quasi-concave; asymmetry
UR - http://eudml.org/doc/250212
ER -

References

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  1. Durante F., Solution of an open problem for associative copulas, Fuzzy Sets and Systems 152 (2005), 411-415. (2005) Zbl1065.03035MR2138520
  2. Genest C., Ghoudi K., Rivest L.-P., Discussion on ``Understanding relationships using copulas'' by E. Frees and E. Valdez, N. Am. Actuar. J. 2 (1999), 143-149. (1999) MR2011244
  3. Klement E.P., Mesiar R., How non-symmetric can a copula be?, Comment. Math. Univ. Carolin. 47 (2006), 141-148. (2006) Zbl1150.62027MR2223973
  4. Klement E.P., Mesiar R., Pap E., Different types of continuity of triangular norms revisited, New Math. Nat. Comput. 1 (2005), 1-17. (2005) Zbl1081.26024MR2158962
  5. Nelsen R.B., An Introduction to Copulas, 2nd edition, Springer, New York, 2006. Zbl1152.62030MR2197664
  6. Nelsen R.B., Extremes of nonexchangeability, Statist. Papers 48 (2007), 329-336. (2007) Zbl1110.62071MR2295821
  7. Robert A.W., Varberg D.E., Convex Functions, Academic Press, New York, 1973. MR0442824

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