| Title:
|
Quasi-concave copulas, asymmetry and transformations (English) |
| Author:
|
Alvoni, Elisabetta |
| Author:
|
Papini, Pier Luigi |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
2 |
| Year:
|
2007 |
| Pages:
|
311-319 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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., {\it Extremes of nonexchangeability\/}, Statist. Papers {\bf 48} (2007), 329--336; Klement E.P., Mesiar R., {\it How non-symmetric can a copula be\/}?, Comment. Math. Univ. Carolin. {\bf 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 $\frac{1}{5}$. 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. (English) |
| Keyword:
|
copula |
| Keyword:
|
quasi-concave |
| Keyword:
|
asymmetry |
| MSC:
|
26B35 |
| MSC:
|
62H05 |
| idZBL:
|
Zbl 1195.62074 |
| idMR:
|
MR2338099 |
| . |
| Date available:
|
2009-05-05T17:03:11Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119661 |
| . |
| Reference:
|
[1] Durante F.: Solution of an open problem for associative copulas.Fuzzy Sets and Systems 152 (2005), 411-415. Zbl 1065.03035, MR 2138520 |
| Reference:
|
[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. MR 2011244 |
| Reference:
|
[3] Klement E.P., Mesiar R.: How non-symmetric can a copula be?.Comment. Math. Univ. Carolin. 47 (2006), 141-148. Zbl 1150.62027, MR 2223973 |
| Reference:
|
[4] Klement E.P., Mesiar R., Pap E.: Different types of continuity of triangular norms revisited.New Math. Nat. Comput. 1 (2005), 1-17. Zbl 1081.26024, MR 2158962 |
| Reference:
|
[5] Nelsen R.B.: An Introduction to Copulas.2nd edition, Springer, New York, 2006. Zbl 1152.62030, MR 2197664 |
| Reference:
|
[6] Nelsen R.B.: Extremes of nonexchangeability.Statist. Papers 48 (2007), 329-336. Zbl 1110.62071, MR 2295821 |
| Reference:
|
[7] Robert A.W., Varberg D.E.: Convex Functions.Academic Press, New York, 1973. MR 0442824 |
| . |
