where where Proof. The proof of Theorem 3 considers sharp bounds on the joint distribution of Y$_{0}$ and Y$_{1}$ only. Sharp bounds on the marginal distributions have been derived in Subsection 3.4 and sharp bounds on the DTE are trivially derived from Lemma 5. Under MTR, it is obvious that F ( y$_{0}$, y$_{1}$ ) = F$_{1}$ ( y$_{1}$ ) for y$_{1}$ ≤ y$_{0}$. Throughout this proof, I consider only the nontrivial case y$_{0}$ < y$_{1}$. To obtain sharp bounds on the joint distribution under M. 1 − M. 5 and MTR, I use the following Lemma B.2 presented by ? . 42 Part 1. Sharp bounds on the joint distribution of Y$_{0}$ and Y$_{1}$ A$_{δ}$ = $^{{}${ a$_{k}$ } ∞ k = $_{−∞}$; 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ for every integer k $^{}}$. F $^{L}$( y$_{0}$, y$_{1}$ ) =          sup z ∈ Ξ   $_{}$max$^{}$     sup y$_{0}$ ≤ y ≤ y$_{1}$      ( P ( y$_{0}$ | 0 , z ) − P ( y | 0 , z )) (1 − p ( z )) + L wst 10 ( y, z )      , 0$^{}$     + max$^{{}$ sup y$_{0}$ ≤ y ≤ y$_{1}$ $^{{}$L mtr 01 ( y$_{0}$, z ) − U wst 01 ( y, z ) + ( P ( Y ≤ y | 1 , z )) p ( z ) $^{}}$, 0$^{}]}$ , if y$_{0}$ < y$_{1}$, F L 1 ( y ) , if y$_{0}$ ≥ y$_{1}$, F U ( y$_{0}$, y$_{1}$ ) =        inf z ∈ Ξ $^{{}$min $^{{}$P ( Y ≤ y$_{0}$ | 0 , z ) (1 − p ( z )) , U mtr 10 ( y, z ) } + min $^{{}$U wst 01 ( y, z ) , P ( y$_{1}$ | 1 , z ) p ( z ) $^{}}}$, if y$_{0}$ < y$_{1}$, F U 1 ( y ) , if y$_{0}$ ≥ y$_{1}$, F U Δ ( δ ) = 1 + inf z ∈ Ξ { p ( z ) + inf y $_{∈$_{R}$}$max $^{{}$P ( y | 1 , z ) p ( z ) − L mtr 01 ( y − δ, z ) , 0 } +inf y $_{∈$_{R}$}$max $^{{}$U mtr 10 ( y, z ) − P ( y − δ | 0 , z ) (1 − p ( z )) , 0 } } , F L $_{Δ}$( δ ) = sup z ∈ Ξ { sup max { a$_{k}$ } ∞ k = $_{−∞}$∈A$_{δ}$ $^{{}$P ( a$_{k}$$_{+1}$ | 1 , z ) p ( z ) − U wst 01 ( a$_{k}$, z ) , 0 } + sup max { b$_{k}$ } ∞ k = $_{−∞}$∈A$_{δ}$ $^{{}$L wst 10 ( b$_{k}$$_{+1}$, z ) − P ( b$_{k}$ | 0 , z ) (1 − p ( z )) , 0 } } ,