in (a) and (b), respectively. Therefore, from consideration of Case a, b and c, where t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w ≤ b$_{k}$$_{+1}$ ≤ b$_{k}$. Consequently, the sharp upper bound is written as follows: letting F U Δ ,W ( δ | w ) be the sharp upper bound on Pr ( Y$_{1}$ − Y$_{0}$ ≤ δ | W = w ) , where t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ δ + w ≤ b$_{k}$$_{+1}$ ≤ b$_{k}$. ■ Appendix B presents the procedure used to compute the sharp lower bound under MTR in Section 4 and Section 5. The following lemma is useful for reducing computational costs: Lemma B.1 Let It is innocuous to assume that { a$_{k}$ } ∞ k = −∞ satisfies a$_{k}$$_{+2}$ − a$_{k}$ > δ for each integer k . Proof. I will show that for any sequence { a$_{k}$ } ∞ k = −∞ ∈ A$_{δ}$ satisfying a$_{k}$$_{+2}$ − a$_{k}$ ≤ δ for some integer k, one can construct {$_{˜}$ a$_{k}$ } ∞ k = −∞ ∈ A$_{δ}$ with ˜ a$_{k}$$_{+2}$ −$_{˜}$ a$_{k}$ > δ for each integer k and 72 ∞ ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } ≤ ∞ ∑ k = −∞ max { F$_{1}$ ($_{˜}$ a$_{k}$$_{+1}$ ) − F$_{1}$ ($_{˜}$ a$_{k}$ ) , 0 } . { a$_{k}$ } ∞ k = −∞ ∈ arg max { a$_{k}$ } ∞ k = $_{−∞}$∈A$_{δ}$ ∞ ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } , where A$_{δ}$ = $^{{}${ a$_{k}$ } ∞ k = $_{−∞}$; 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ for each integer k $^{}}$. F U Δ ( δ ) = ∫ F U Δ ,W ( δ | w ) dF$_{W}$ ( w ) = ∫ { 1 − sup { B$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max $^{{}$µ$_{0}$$_{,W}$ ( B$_{k}$ | w ) − µ$_{1}$$_{,W}$ $^{(}$B D k | w $^{)}$, 0 } } dF$_{W}$ = 1 +$^{∫}$ inf { b$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ min$^{{}$ F$_{1}$$_{,W}$ $^{(}$t$_{1}$ − t$_{W}$ t$_{0}$ − t$_{W}$ b$_{k}$$_{+1}$ − t$_{1}$ − t$_{0}$ t$_{0}$ − t$_{W}$ w | w$^{)}$ − F$_{0}$$_{,W}$ ( b$_{k}$ | w ) , 0$^{}}$ dF$_{W}$ sup { B$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max $^{{}$µ$_{0}$$_{,W}$ ( B$_{k}$ | w ) − µ$_{1}$$_{,W}$ $^{(}$B D k | w $^{)}$, 0 } = sup { b$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max$^{{}$ F$_{0}$$_{,W}$ ( b$_{k}$ | w ) − F$_{1}$$_{,W}$ $^{(}$t$_{1}$ − t$_{W}$ t$_{0}$ − t$_{W}$ b$_{k}$$_{+1}$ − t$_{1}$ − t$_{0}$ t$_{0}$ − t$_{W}$ w | w$^{)}$ , 0$^{}}$ Appendix B