content_image/315912.jpg content_image/315916.jpg content_image/315917.jpg content_image/315918.jpg content_image/315919.jpg content_image/315922.jpg
and these bounds are sharp. 

<img src='content_image/315912.jpg'>

From Lemma 3, sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ are improved based on L mtr 01 ( y, z ) and U mtr 10 ( y | z ) under M. 1 − M. 4, and MTR as follows: 

<img src='content_image/315916.jpg'>

<img src='content_image/315917.jpg'>

<img src='content_image/315918.jpg'>

<img src='content_image/315919.jpg'>

Now, I show that MTR also has identifying power for the joint distribution. I will use Lemma 4 to bound the joint distribution under MTR. Henceforth, x + denotes max ( x, 0) . 

Lemma 4 ( ? ) Suppose that marginal distributions F$_{0}$ and F$_{1}$ are known and that F ( a$_{0}$, a$_{1}$ ) = θ where ( a$_{0}$, a$_{1}$ ) ∈ R 2 and θ satisfies max ( F$_{0}$ ( a$_{0}$ ) + F$_{1}$ ( a$_{1}$ ) − 1 , 0) ≤ θ ≤ min ( F$_{0}$ ( a$_{0}$ ) , F$_{1}$ ( a$_{1}$ )) . Then, sharp bounds on the joint distribution F are given as follows: 

<img src='content_image/315922.jpg'>

