content_image/318186.jpg content_image/318185.jpg content_image/318184.jpg
## 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. 

## Example 1 (Continued). 

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 

<img src='content_image/318186.jpg'>

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 

<img src='content_image/318185.jpg'>

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 

<img src='content_image/318184.jpg'>

