Figure 3: { Y$_{0}$ ∈ A$_{0}$, Y$_{1}$ ∈ A$_{1}$ } Graphically, the DTE corresponds to the region below the straight line Y$_{1}$ = Y$_{0}$ + δ in the support space as shown in Figure 4. Since the given marginal distributions are informative on the joint distribu- tion for rectangular regions in the support space, one can bound the DTE by considering two rectangles { Y$_{0}$ ≥ y − δ, Y$_{1}$ ≤ y } and { Y$_{0}$ < y ′ − δ, Y$_{1}$ > y $^{′}$} for any ( y, y $^{′}$) ∈$_{R}$ $^{2}$. Although the probability of each rect- angle is not point-identified, it can be bounded by Fr´echet inequalities. 7 Since the DTE is bounded from below by the Fr´echet lower bound on Pr { Y$_{0}$ ≥ y − δ, Y$_{1}$ ≤ y } for any y ∈$_{R}$ , the lower bound on the DTE is obtained as follows: Similarly, the DTE is bounded from above by 1 − Pr { Y$_{0}$ < y ′ − δ, Y$_{1}$ > y $^{′}$} for any y ′ ∈$_{R}$ . Therefore, the upper bound on the DTE is obtained by the Fr´echet lower bound on Pr { Y$_{0}$ < y ′ − δ, Y$_{1}$ > y $^{′}$} as follows: Makarov (1981) proved that those lower and upper bounds are sharp. 8 If the marginal distributions of Y$_{0}$ and Y$_{1}$ are both absolutely continuous with respect to the Lebesgue measure on$_{R}$, then the Makarov upper bound and lower bound are achieved when F ( y$_{0}$, y$_{1}$ ) = C L s ( F$_{0}$ ( y$_{0}$ ) , F$_{1}$ ( y$_{1}$ )) $^{8}$One may wonder if multiple rectangles below Y$_{1}$ = Y$_{0}$ + δ that overlap one another could yield the more improved lower bound. However, if the Fr´echet lower bound on another rectangle { Y$_{0}$ ≥ y ′′ − δ, Y$_{1}$ ≤ y $^{′′}$} is added and the Fr´echet upper bound on the intersection of the two rectangles is subtracted, it is smaller than or equal to the lower bound obtained from the only one rectangle. 10 sup y max ( F$_{1}$ ( y ) − F$_{0}$ ( y − δ ) , 0) ≤ F$_{Δ}$ ( δ ) . F$_{Δ}$ ( δ ) ≤ 1 − sup y max ( F$_{0}$ ( y − δ ) − F$_{1}$ ( y ) , 0) . $^{7}$Note that Fr´echet lower bounds on Pr { Y$_{0}$ ≥ y ′ − δ, Y$_{1}$ ≤ y $^{′}$} and Pr { Y$_{0}$ < y ′ − δ, Y$_{1}$ > y $^{′}$} are sharp. They are both achieved when Y$_{0}$ and Y$_{1}$ are perfectly positively dependent.