Figure 4: Makarov bounds and when F ( y$_{0}$, y$_{1}$ ) = C U t ( F$_{0}$ ( y$_{0}$ ) , F$_{1}$ ( y$_{1}$ )) respectively, where Note that both C U s ( u, v ) and C L t ( u, v ) depend on δ , through s and t , respectively. 9 Since the joint distri- bution achieving Makarov bounds varies with δ, Makarov bounds are only pointwise sharp, not uniformly . To address this issue, Firpo and Ridder (2008) proposed joint bounds on the DTE for multiple values of δ , which are tighter than Makarov bounds. However, their improved bounds are not sharp and sharp bounds on the functional F$_{Δ}$ are an open question. For details, see Frank et al. (1997), Nelsen (2006) and Firpo and Ridder (2008). Although Makarov bounds are sharp when no other restrictions are imposed, they are often too wide to be informative in practice and not sharp in the presence of additional restrictions on the set of possible pairs of potential outcomes. Figure 5 illustrates that if the support is restricted to the region above the straight line Y$_{1}$ = Y$_{0}$ by MTR, the Makarov lower bound is not the best possible anymore. The lower bound can be $^{9}$To be precise, when the distribution of Y$_{1}$ − Y$_{0}$ is discontinuous, the Makarov lower bound is attained only for the left limit of the DTE. That is, F$_{Δ}$ $^{(}$δ − ) = F L Δ $^{(}$δ − ) = t under C L t , while under C U s , F$_{Δ}$ ( δ ) = F U Δ ( δ ) = s for the right-continuous distribution function F$_{Δ}$ . Note that even if both marginal distributions of Y$_{1}$ and Y$_{0}$ are continuous, the distribution of Y$_{1}$ − Y$_{0}$ may not be continuous. Hence, typically the lower bound on the DTE is established only for the left limit of the DTE Pr [ Y$_{1}$ − Y$_{0}$ < δ ] . See Nelsen (2006) for details. 11 s = F U Δ ( δ ) and t = F L Δ $^{(}$δ − $^{)}$, C U s ( u, v ) =      min ( u + s − 1 , v ) , 1 − s ≤ u ≤ 1 , 0 ≤ v ≤ s, max ( u + v − 1 , 0) , elsewhere, C L t ( u, v ) =      min ( u, v − t ) , 0 ≤ u ≤ 1 − t, t ≤ v ≤ 1 , max ( u + v − 1 , 0) , elsewhere.