content_image/299313.jpg content_image/299316.jpg content_image/299312.jpg
given by 

<img src='content_image/299313.jpg'>

where we use the centered extension of D Ψ 2 as discussed in Section 2.3. Here and throughout, P$_{n}$ denotes the empirical distribution. It is straightforward to verify that indeed ψ$_{n}$ = ψ$_{n,NR}$ . 

## 3.2 Inference under the null 

## 3.2.1 Asymptotic behavior 

For each P ∈ M , we let ˜ Γ$_{P}$ be the P$_{0}$ -centered modification of Γ$_{P}$ given by 

<img src='content_image/299316.jpg'>

and denote ˜ Γ$_{P}$$_{0}$ by ˜ Γ$_{0}$. While ˜ Γ$_{0}$ = Γ$_{0}$ under H$_{0}$ , this is not true more generally. Below, we use Rem Ψ n and ˜ Γ$_{n}$ to respectively denote Rem Ψ P and ˜ Γ$_{P}$ evaluated at P = ˆ P$_{n}$ . Straightforward algebraic manipulations allows us to write 

<img src='content_image/299312.jpg'>

Our objective is to show that n ( ψ$_{n}$ − ψ$_{0}$ ) behaves like n$_{U}$$_{n}$ Γ$_{0}$ as n gets large under H$_{0}$ . In view of (4), this will be true, for example, under conditions ensuring that C1) n ( P$_{n}$ − P$_{0}$ ) P$_{0}$ Γ$_{n}$ = o$_{P}$$_{0}$ (1) (empirical process and consistency conditions); C2) n$_{U}$$_{n}$ $^{(}$˜ Γ$_{n}$ − Γ$_{0}$ ) = o$_{P}$$_{0}$ (1) ( U -process and consistency conditions); C3) n Rem Ψ n = o$_{P}$$_{0}$ (1) (consistency and rate conditions). 

We have already argued that C3) is reasonable in many examples of interest, including those presented in this paper. Nolan and Pollard (1987, 1988) developed a formal theory that 

