Figure 9: a$_{k}$$_{+1}$ ≤ a$_{k}$ + δ at the optimum that a$_{k}$$_{+1}$ > a$_{k}$ + δ for some integer k . This implies that triangles in the region between two straight lines Y$_{1}$ = Y$_{0}$ + δ and Y$_{1}$ = Y$_{0}$ lie sparsely as shown in Figure 9(a). Then by adding extra triangles that fill the empty region between two sparse triangles as shown in Figure 9(b), one can always construct a sequence of mutually exclusive triangles that yield the identical or improved lower bound. Therefore, without loss of generality, one can assume a$_{k}$$_{+1}$ ≤ a$_{k}$ + δ for every integer k . On the other hand, ones cannot exclude the case where a$_{k}$$_{+1}$ < a$_{k}$ + δ for some integer k at the optimum . This implies that for some k, the triangle is not large enough to fit in the region corresponding to the DTE under MTR as shown in Figure 10(b). It depends on the underlying joint distribution which sequence of triangles would yield the tighter lower bound, and it is possible that a$_{k}$$_{+1}$ < a$_{k}$ + δ for some integer k at the optimum . Therefore, Consequently, for δ ≥ 0 , where 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ. 23 A D k = ( a$_{k}$ + δ, ∞ ) ∪ ( a$_{k}$$_{+1}$, ∞ ) = (min { a$_{k}$ + δ, a$_{k}$$_{+1}$ } , ∞ ) = ( a$_{k}$$_{+1}$, ∞ ) . F L Δ ( δ ) = sup { A$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max $^{{}$µ$_{0}$ ( A$_{k}$ ) − µ$_{1}$ $^{(}$A D k $^{)}$, 0 } = sup { a$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 }