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under NSM, for any ̂ p ( z ) ∈ ( p ( z ) , 1] , 

<img src='content_image/302425.jpg'>

Since P $^{{}$ε$_{1}$ ≤ m − $^{1}$(1 , y ) | p ( z ) < U ≤ ̂ p ( z ) $^{}}$is nondecreasing in ̂ p ( z ) by NSM, for z ∈ Z \ p − $^{1}$( p ) , the highest possible observable lower bound is obtained when ̂ p ( z ) = p . Therefore by NSM, for any z ∈ Z \ p − $^{1}$( p ) , NSM implies 

<img src='content_image/302426.jpg'>

Obviously, P $^{(}$ε$_{1}$ ≤ m − $^{1}$(1 , y ) , U ≤ p ) and P $^{(}$ε$_{1}$ ≤ m − $^{1}$(1 , y ) , U ≤ p ( z ) ) are point-identified as lim p ( z ) → p P ( y, 1 | z ) and P ( y, 1 | z ) for any z ∈ Z . 

Similarly, P$_{0}$ ( y | 1 , z ) = P $^{(}$ε$_{0}$ ≤ m − $^{1}$(0 , y ) | U ≤ p ( z ) ) and by NSM, for any z ∈ Z \ p − 1 $^{(}$p) 

<img src='content_image/302427.jpg'>

Also, P $^{(}$ε$_{0}$ ≤ m − $^{1}$(0 , y ) , p < U $^{)}$and P $^{(}$ε$_{0}$ ≤ m − $^{1}$(0 , y ) , p ( z ) < U $^{)}$are point-identified as lim p ( z ) → p P ( y, 0 | z ) and P ( y, 0 | z ), respectively, for any z ∈ Z . These bounds are tighter than bounds obtained without NSM. 

On the other hand, NSM has no additional identifying power on the upper bound on P$_{1}$ ( y | 0 , z ) and the lower bound on P$_{0}$ ( y | 1 , z ) , which means that these bounds under NSM are identical to those obtained without NSM . 

