Figure 4: Improved lower bound on the DTE under MTR where Suppose that marginal distributions F$_{0}$ and F$_{1}$ are fixed. Lemma 4 shows that sharp bounds on the joint distribution improve when the values of the joint distribution are known at some fixed points. Note that P ( Y$_{1}$ ≥ Y$_{0}$ ) = 1 if and only if F ( y, y ) = F$_{1}$ ( y ) for all y ∈ $_{R}$. As illustrated in Figure 3, Therefore, Since for each y ∈ R the value of F ( y, y ) is known from the fixed marginal distribution F$_{1}$ under MTR, sharp bounds on the joint distribution can be derived by taking the intersection of the bounds under the restriction F ( y, y ) = F$_{1}$ ( y ) over all y ∈$_{R}$ . Technical details are presented in Appendix. 22 P ( Y$_{0}$ > Y$_{1}$ ) = P [ ∪ y ∈$_{R}$ { Y$_{0}$ > y, Y$_{1}$ < y }$^{]}$ . P ( Y$_{1}$ ≥ Y$_{0}$ ) = 1 ⇐⇒ P ( Y$_{0}$ > Y$_{1}$ ) = 0 ⇐⇒ P ( Y$_{0}$ > y, Y$_{1}$ < y ) = 0 for all y ∈$_{R}$ ⇐⇒ F ( y, y ) = F$_{1}$ ( y ) , for all y ∈$_{R}$ . F $^{L}$( y$_{0}$, y$_{1}$ ) = max $^{{}$0 , F$_{0}$ ( a$_{0}$ ) + F$_{1}$ ( a$_{1}$ ) − 1 , θ − ( F$_{0}$ ( a$_{0}$ ) − F$_{0}$ ( y$_{0}$ )) + − ( F$_{1}$ ( a$_{1}$ ) − F$_{1}$ ( y$_{1}$ )) + $^{}}$, F $^{L}$( y$_{0}$, y$_{1}$ ) = min $^{{}$F$_{0}$ ( y$_{0}$ ) , F$_{1}$ ( y$_{1}$ ) , θ + ( F$_{0}$ ( y$_{0}$ ) − F$_{0}$ ( a$_{0}$ )) + + ( F$_{1}$ ( y$_{1}$ ) − F$_{1}$ ( a$_{1}$ )) + $^{}}$.