content_image/296494.jpg content_image/296495.jpg content_image/296496.jpg
Now to derive the lower bound F $^{L}$( y$_{0}$, y$_{1}$ ) , let G ( y ) = F$_{1}$ ( y ) − ( F$_{0}$ ( y ) − F$_{0}$ ( y$_{0}$ )) $^{+}$− ( F$_{1}$ ( y ) − F$_{1}$ ( y$_{1}$ )) $^{+}$. Then for y$_{0}$ < y$_{1}$$_{,}$ 

and so, 

<img src='content_image/296494.jpg'>

<img src='content_image/296495.jpg'>

Since F$_{1}$ ( y$_{1}$ ) − F$_{0}$ ( y$_{1}$ ) + F$_{0}$ ( y$_{0}$ ) ≥ max (0 , F$_{0}$ ( y$_{0}$ ) + F$_{1}$ ( y$_{1}$ ) − 1) , for y$_{0}$ < y$_{1}$, 

## Corollary 2 

<img src='content_image/296496.jpg'>

Corollary 2 (Bounds on the marginal distributions of potential outcomes) Under M. 1 − M. 4 , PSM and MTR, sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ , their joint distribution and the 

