content_image/308956.jpg content_image/308955.jpg content_image/308954.jpg
The upper bound is obtained as Fr´echet-Hoeffing upper bound as follows: 

<img src='content_image/308956.jpg'>

The lower bound is obtained when ε$_{0}$ and ε$_{1}$ are independent conditionally on U , while the upper bound is obtained when ε$_{0}$ and ε$_{1}$ are perfectly dependent conditionally on U . Thus they are sharp. 

## Part 2. Sharp bounds on the DTE 

To show that CPQD has no additional identifying power on the DTE, I use the following Lemma which has been presented by ? and ? . 

Lemma B.1 Let C denote a lower bound on the copula of X and Y , and F$_{X}$$_{+}$$_{Y}$ denote the distribu- tion function of X + Y. If support of ( X, Y ) , supp ( X, Y ) satisfies supp ( X, Y ) = supp ( X ) × supp ( Y ) , 

where C $^{d}$( u, v ) = u + v − C ( u, v ) . 

<img src='content_image/308955.jpg'>

Let Y$_{1}$ = X and Y$_{0}$ = − Y . By Lemma B.1, sharp bounds on the DTE are affected by only the upper bound on the copula of Y$_{0}$ and Y$_{1}$. Since CPQD improves only the lower bound on the copula if Y$_{0}$ and Y$_{1}$ , the DTE bounds do not improve by CPQD. 

## Theorem 3 

Theorem 3 Under M. 1 − M. 4 and MTR, sharp bounds on F ( y$_{0}$, y$_{1}$ ) , and F$_{Δ}$ ( δ ) are given as follows: for d ∈ { 0 , 1 } , y ∈$_{R}$ , δ ∈$_{R}$ , and ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ ×$_{R}$ , 

<img src='content_image/308954.jpg'>

