Figure 11: The DTE under concave/convex treatment response where with (ii) Under convex treatment response, 25 F L $_{Δ}$( δ ) = ∫ sup { a$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max { F$_{1}$$_{,W}$ ( S$_{1}$a$_{k}$$_{+1}$ + (1 − S$_{1}$ ) w | w ) − F$_{0}$$_{,W}$ ( a$_{k}$ | w ) , 0 } dF$_{W}$ , F U Δ ( δ ) = 1 +$^{∫}$ inf y ∈$_{R}$ { min ( F$_{1}$$_{,W}$ ( y | w ) − F$_{0}$$_{,W}$ ( y − δ | w )) , 0 } dF$_{W}$ . 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ, T$_{0}$ ( b$_{k}$ + δ ) + T$_{1}$ ≤ b$_{k}$$_{+1}$ ≤ b$_{k}$, where T$_{1}$ = t$_{1}$ − t$_{0}$ t$_{1}$ − t$_{W}$ , T$_{0}$ = 1 − T$_{1}$. F L $_{Δ}$( δ ) = sup { a$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ ∫ max { F$_{1}$$_{,W}$ ( a$_{k}$$_{+1}$ | w ) − F$_{0}$$_{,W}$ ( a$_{k}$ | w ) , 0 } dF$_{W}$ , F U Δ ( δ ) = 1 +$^{∫}$ inf { b$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ { min$^{(}$ F$_{1}$$_{,W}$ ( 1 T$_{0}$ b$_{k}$$_{+1}$ − T$_{1}$ T$_{0}$ w | w$^{)}$ − F$_{0}$$_{,W}$ ( b$_{k}$ | w )$^{)}$ , 0$^{}}$ dF$_{W}$ ,