Figure 7: True DTE and bounds on the DTE consisting of the lower bound. Also, as discussed in Chapter 1, the sharp lower bound on F$_{Δ}$ ( δ ) under MTR converges to the Makarov lower bound as δ increases for sufficiently large values of δ. On the other hand, the upper bound under NSM does not improve from the worst upper bound as discussed in Subsection 3.2 Although the upper bound improves under MTR through improvement in the lower bound on P$_{0}$ ( y, 1 | z ) and the upper bound on P$_{1}$ ( y, 0 | z ), the improvement in the upper bound under MTR is not remarkable as shown in Figure 7. Also, the quantiles of treatment effects can be obtained by inverting the bounds on the DTE. The bounds on the quantiles of the DTE are reported in Table 1. Table 2 shows the bounds on the joint distribution under various restrictions considered in this study. Compared to the worst bounds, bounds are tighter under NSM due to the marginal distributions bounds improved by NSM. On the other hand, the upper bound under CQPD does not improve unlike the lower bound. Note that CQPD has no identifying power on marginal dis- tributions, while it improves the lower bound on the joint distribution. However, when CQPD is combined with NSM, the upper bound also improves due to the improved marginal distributions bounds under NSM. The identification region under MTR is tighter than the worst identification region for both the upper bound and the lower bound. Note that the upper bound under MTR 28