As is well--known, the Fr\'{e}chet--Hoeffding bounds are the best--possible for both copulas andquasi--copulas: for every (quasi--)copula $Q$, $\max\{x+y-1,0\}\le Q(x,y)\le \min\{x,y\}$ for all $x,y\in\uint$. Sharper bounds hold when the (quasi--)copulas take prescribed values, \eg, along their diagonal or horizontal resp.\ vertical sections. Here we pursue two goals: first, we investigate construction methods for (quasi--)copulas with a given sub--diagonal section, \ie, with prescribed values along the straight line segment joining the points $(x_0,0)$ and $(1,1-x_0)$ for $x_0\in\opint{0,1}$. Then, we determine the best--possible bounds for sets of quasi--copulas with a given sub--diagonal section.
Quasi--copulas with a given sub--diagonal section
SEMPI, Carlo
2008-01-01
Abstract
As is well--known, the Fr\'{e}chet--Hoeffding bounds are the best--possible for both copulas andquasi--copulas: for every (quasi--)copula $Q$, $\max\{x+y-1,0\}\le Q(x,y)\le \min\{x,y\}$ for all $x,y\in\uint$. Sharper bounds hold when the (quasi--)copulas take prescribed values, \eg, along their diagonal or horizontal resp.\ vertical sections. Here we pursue two goals: first, we investigate construction methods for (quasi--)copulas with a given sub--diagonal section, \ie, with prescribed values along the straight line segment joining the points $(x_0,0)$ and $(1,1-x_0)$ for $x_0\in\opint{0,1}$. Then, we determine the best--possible bounds for sets of quasi--copulas with a given sub--diagonal section.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
