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Title: On bivariate copula mappings Authors:  Gediminas Bagdonas - Vilnius University (Lithuania)
Martynas Manstavicius - Vilnius University (Lithuania) [presenting]
Abstract: Inspired by many examples from the literature, we are concerned with a particular method of bivariate copula construction, namely, for a given copula $C:[0,1]^2\to[0,1]$ and a function $f:[0,1]\to\mathbb{R}_+$ we let $H_f(C)(x,y):=C(x,y)f(\overline{C}(x,y))$, where $\overline{C}(x,y):=1-x-y+C(x,y)$, $(x,y)\in[0,1]^2$, is the survival function associated with copula $C$. To classify the functions $f$ based on the properties of $H_f(\cdot)$, we call a particular function $f$ {\it eligible} if $H_f(C)$ is a copula for any bivariate copula $C$, {\it conditionally-eligible} if $H_f(C_1)$ is a copula but $H_f(C_2)$ is not a copula for some bivariate copulas $C_1, C_2$, and {\it non-eligible} if $H_f(C)$ is not a copula for any bivariate copula $C$. We then provide necessary and sufficient conditions for $f$ to be eligible, and illustrate with examples. Some of the statistical properties of $H_f(C)$ for eligible $f$ are also discussed.