content_image/306436.jpg content_image/306435.jpg
Therefore, sharp bounds on the DTE are obtained from the intersection bounds as follows: 

## Corollary 1 

<img src='content_image/306436.jpg'>

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$_{1}$ ) ∈$_{R}$ ×$_{R}$ , 

<img src='content_image/306435.jpg'>

