Theorem 1 Part 1. Sharp bounds on marginal distributions F$_{0}$ ( · ) and F$_{1}$ ( · ) Proof. The proof consists of three parts: sharp bounds on (i) marginal distributions, (ii) the joint distribution, and (iii) the DTE. where Theorem 1 Under M. 1 − M. 4 , sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ , their joint distribution and the DTE are obtained as follows: for d ∈ { 0 , 1 } , y ∈$_{R}$ , δ ∈$_{R}$ , and ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ ×$_{R}$ , 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 ) $^{]}$, F L 0 ( y ) = sup z ∈ Ξ $^{[}$P { y | 0 , z } (1 − p ( z )) + L wst 01 ( y, z ) $^{]}$, (11) F U 0 ( y ) = inf z ∈ Ξ $^{[}$P { y | 0 , z } (1 − p ( z )) + U wst 01 ( y, z ) $^{]}$, F L 1 ( y ) = sup z ∈ Ξ $^{[}$P { y | 1 , z } p ( z ) + L wst 10 ( y, z ) $^{]}$, F U 1 ( y ) = inf z ∈ Ξ $^{[}$P { y | 1 , z } p ( z ) + U wst 10 ( y, z ) $^{]}$, F $^{L}$( y$_{0}$, y$_{1}$ ) = sup z ∈ Ξ    max $^{{}$( P ( y$_{0}$ | 0 , z ) − 1) (1 − p ( z )) + L wst 10 ( y$_{1}$, z ) , 0 } + max $^{{}$L wst 01 ( y$_{0}$, z ) + ( P ( y$_{1}$ | 1 , z ) − 1) p ( z ) , 0 }   $_{}$, F U ( y$_{0}$, y$_{1}$ ) = inf z ∈ Ξ    min $^{{}$P ( y$_{0}$ | 0 , z ) (1 − p ( z )) , U wst 10 ( y$_{1}$, z ) } + min $^{{}$U wst 01 ( y$_{0}$, z ) , P ( y$_{1}$ | 1 , z ) p ( z ) }   $_{}$, F L Δ ( δ ) = sup z ∈ Ξ    sup max y ∈$_{R}$ $^{{}$P ( y | 1 , z ) p ( z ) − U wst 01 ( y − δ, z ) , 0 } +sup max y ∈$_{R}$ $^{{}$L wst 10 ( y, z ) − P ( y − δ | 0 , z ) (1 − p ( z )) , 0 }    , F U Δ ( δ ) = 1 + inf z ∈ Ξ    inf min y ∈$_{R}$ $^{{}$P ( y | 1 , z ) p ( z ) − L wst 01 ( y − δ, z ) , 0 } +inf min y ∈$_{R}$ $^{{}$U wst 10 ( y, z ) − P ( y − δ | 0 , z ) (1 − p ( z )) , 0 }   $_{}$. 34