content_image/295922.jpg content_image/295921.jpg content_image/295920.jpg
M.5 The propensity score p ( z ) is bounded away from 0 and 1. 

Under M. 5, Chebyshev’s integral inequality yields the lower bound as follows: 

<img src='content_image/295922.jpg'>

The inequality in the third line of (7) follows from CPQD and the inequality in the fourth line of (7) is due to Chebyshev’s integral inequality. Consequently, I obtain the following: 

<img src='content_image/295921.jpg'>

Similarly, the lower bound on P ( Y$_{0}$ ≤ y$_{0}$, Y ≤ y$_{1}$ | 1 , z ) is obtained as follows: 

<img src='content_image/295920.jpg'>

Interestingly, the DTE is still bounded by Makarov bounds under CPQD although the lower bound on the joint distribution improves. The rigorous proof is provided in Appendix. Here I discuss the reason intuitively using a graphical illustration. As shown in Figure 1, the DTE is a probability corresponding to the region below the straight line y$_{1}$ = y$_{0}$ + δ and the Makarov lower bound is obtained from the rectangle { Y$_{0}$ ≥ y − δ, Y$_{1}$ ≤ y } below the straight line Y$_{1}$ = Y$_{0}$ + δ for y ∈ R that maximizes the Fr´ echet-Hoeffding lower bound. Since the Fr´ echet-Hoeffding lower bound on P ( Y$_{0}$ ≥ y − δ, Y$_{1}$ ≤ y ) for each y ∈$_{R}$ is achieved when the joint distribution of Y$_{0}$ and 

