content_image/303692.jpg content_image/303691.jpg content_image/303690.jpg content_image/303689.jpg
are obtained by Fr´ echet-Hoeffding bounds as follows: for any ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ $^{2}$, 

<img src='content_image/303692.jpg'>

Since P$_{1}$ ( y$_{1}$ | 0 , z ) is only partially identified, sharp bounds on P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) are ob- tained by taking the union over all possible values of P$_{1}$ ( y$_{1}$ | 0 , z ) . Therefore, sharp bounds on P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$, D = 0 | z ) = P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) (1 − p ( z )) are derived as follows: 

Similarly, 

<img src='content_image/303691.jpg'>

<img src='content_image/303690.jpg'>

By (12), sharp bounds on P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ ) are obtained by taking the intersection of the bounds over all values of z ∈ Ξ , 

<img src='content_image/303689.jpg'>

