Therefore, sharp bounds on the DTE are obtained from the intersection bounds as follows: Corollary 1 (Bounds on the marginal distributions of potential outcomes) Under M. 1 − M. 4 and SM, sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ , their joint distribution and the DTE are given as follows: for d ∈ { 0 , 1 } , y ∈$_{R}$ , δ ∈$_{R}$ , and ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ ×$_{R}$ , 38 F$_{d}$ ( y ) ∈ $^{[}$F L d ( y ) , F U d ( y ) $^{]}$, F ( y$_{0}$, y$_{1}$ ) ∈ $^{[}$F $^{L}$( y$_{0}$, y$_{1}$ ) , F U ( y$_{0}$, y$_{1}$ ) $^{]}$, F$_{Δ}$ ( δ ) ∈ $^{[}$F L Δ ( δ ) , F U Δ ( δ ) $^{]}$, sup z ∈ Ξ { sup max y ∈$_{R}$ $^{{}$L wst 10 ( y, z ) − P ( y − δ | 0 , z ) (1 − p ( z )) , 0 } +sup max y ∈$_{R}$ $^{{}$P ( y | 1 , z ) p ( z ) − U wst 01 ( y − δ | z ) , 0 } } ≤ P ( Y$_{1}$ − Y$_{0}$ ≤ δ ) ≤ 1 + inf z ∈ Ξ { inf max y ∈$_{R}$ $^{{}$P ( y | 1 , z ) p ( z ) − L wst 01 ( y − δ, z ) , 0 } +inf max y ∈$_{R}$ $^{{}$U wst 10 ( y, z ) − P ( y − δ | 0 , z ) (1 − p ( z )) , 0 } . Corollary 1