5 Illustrations We now return to each of our examples. We first show that Examples 1, 2 and 3 satisfy the regularity conditions described in Section 2. Specifically, we show that all involved parame- ters R and S belong to S under reasonable conditions. Furthermore, we determine explicit remainder terms for the asymptotic representation used in each example and describe con- ditions under which these remainder terms are negligible. For any T ∈ S , we will use the shorthand notation ˙ T$_{˜}$ $_{t}$( x $^{T}$) ≜ d $_{dt}$T$_{P}$$_{t}$ ( x $^{T}$) ∣ t = ˜ t for ˜ t in a neighborhood of zero. The parameter S with S$_{P}$ ≡ 0 belongs to S trivially, with D S P ≡ 0. Condition (S1) holds with x $^{R}$( o ) = w . Condition (S2) holds using that R$_{t}$ ( w ) equals Since we must only consider h$_{1}$ and h$_{2}$ uniformly bounded, for t sufficiently small, we see that R$_{t}$ ( w ) is twice continuously differentiable with uniformly bounded derivatives. Condi- tion (S3) is satisfied by and D S P ≡ 0. If min$_{a}$ P ( A = a | W ) is bounded away from zero with probability 1 uniformly in P , it follows that ( P,o ) ↦→ D R $_{P}$( o ) is uniformly bounded. Clearly, we have that Rem S P ≡ 0. We can also verify that Rem R $_{P}$( o ) equals 23 1 ∑ ˜ a =0 ( − 1) ˜ $^{a}$E$_{P}$$_{0}$ { [ 1 − P$_{0}$ ( A = ˜ a | W ) P ( A = ˜ a | W ) ] × [ E$_{P}$ ( Y | A,W ) − E$_{P}$$_{0}$ ( Y | A,W )] ∣ $_{∣}$A = ˜ a,W = w $^{}}$. D R $_{P}$( o ) ≜ 2 a − 1 P ( A = a | W = w ) { y − E$_{P}$ [ Y | A = a,W = w ] } 1 ∑ a =0 ( − 1) a +1 ∫ y$^{{}$ 1 + th$_{1}$ ( w,a,y ) + t $^{2}$h$_{2}$ ( w,a,y ) 1 + tE$_{P}$$_{0}$ [ h$_{1}$ ( w,A,Y )] + t $^{2}$E$_{P}$$_{0}$ [ h$_{2}$ ( w,A,Y )]$^{}}$ dP$_{0}$ ( y | a,w ) . (8) Example 1 (Continued).