Exact upper bound for copulas with a given diagonal section

Abstract

We answer a 15-year-old open question about the exact upper bound for bivariate copulas with a given diagonal section by giving an explicit formula for this bound. As an application, we determine the maximal asymmetry of bivariate copulas with a given diagonal section and construct a copula that attains it. We derive a formula for the maximal asymmetry that is simple enough to be used by practitioners.

Figures (7)

• Figure 1: Two subsequences of the sequence $x=x_0 < x_1 < \ldots < x_n = y$ approximating the total variation of $\widehat{\delta}$ on $[x,y]$ in the proof of Proposition \ref{['pr:bound']}.
• Figure 2: Disjoint unions of boxes used in the derivation of inequalities \ref{['eq:red']} and \ref{['eq:green']}.
• Figure 3: The graph of copula $U_\delta$ with $\delta(x)=x^2$ (left) and its scatterplot with the corresponding regions $D_f(\delta)$, $D_x(\delta)$, and $D_y(\delta)$ (right).
• Figure 4: The graphs of functions $K_\delta$ (left), $\overline{C}_\delta$ (middle), and $A_\delta$ (right) with $\delta$ from Example \ref{['ex:KCA']}.
• Figure 5: Upright versus lying shaded rectangle representing the condition $(ii)$ of Theorem \ref{['th:CequalA']} - in the first case we have $\overline{C}_\delta \neq A_\delta$, while in the second we have $\overline{C}_\delta=A_\delta$.

Theorems & Definitions (41)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6: NeLaUbFl04FrNe97
• Theorem 2.7: FrNe02
• Definition 2.8
• Proposition 3.1
• Remark 3.2