content_image/315544.jpg content_image/315543.jpg
where 

<img src='content_image/315544.jpg'>

where 

<img src='content_image/315543.jpg'>

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. 

## Part 1. Sharp bounds on the joint distribution of Y$_{0}$ and Y$_{1}$ 

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 ? . 

