Lemma B.2 Let C be a copula, and suppose C ( a, b ) = θ, where ( a, b ) is in (0 , 1) 2 and θ sat- isfies max ( a + b − 1 , 0) ≤ θ ≤ min ( a, b ). Then where C$_{U}$ and C$_{L}$ are the copulas given by where ( x ) + = max { x, 0 } . Lemma B.3 For fixed marginal distribution functions F$_{0}$ and F$_{1}$, sharp bounds on the joint distribution function F are given as follows: where 43 F $^{L}$( y$_{0}$, y$_{1}$ ) = max y$_{0}$ ≤ $_{y<y$_{1}$}${ F$_{1}$ ( y ) − F$_{0}$ ( y ) + F$_{0}$ ( y$_{0}$ ) } , F U ( y$_{0}$, y$_{1}$ ) = inf y $_{∈$_{R}$}$min ( F$_{0}$ ( y$_{0}$ ) , F$_{1}$ ( y$_{1}$ )) . F $^{L}$( y$_{0}$, y$_{1}$ ) ≤ F ( y$_{0}$, y$_{1}$ ) ≤ F U ( y$_{0}$, y$_{1}$ ) C$_{U}$ ( u, v ) = min $^{(}$u, v, θ + ( u − a ) + + ( v − b ) + $^{)}$, C$_{L}$ ( u, v ) = max $^{(}$0 , u + v − 1 , θ − ( a − u ) + − ( b − v ) + $^{)}$. C$_{L}$ ( u, v ) ≤ C ( u, v ) ≤ C$_{U}$ ( u, v ) ,