given by 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$_{n,NR}$ . For each P ∈ M , we let ˜ Γ$_{P}$ be the P$_{0}$ -centered modification of Γ$_{P}$ given by and denote ˜ Γ$_{P}$$_{0}$ by ˜ Γ$_{0}$. While ˜ Γ$_{0}$ = Γ$_{0}$ under H$_{0}$ , this is not true more generally. Below, we use Rem Ψ n and ˜ Γ$_{n}$ to respectively denote Rem Ψ P and ˜ Γ$_{P}$ evaluated at P = ˆ P$_{n}$ . Straightforward algebraic manipulations allows us to write Our objective is to show that n ( ψ$_{n}$ − ψ$_{0}$ ) behaves like n$_{U}$$_{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 15 ψ$_{n}$ − ψ$_{0}$ = $_{U$_{n}$}$Γ$_{n}$ − ψ$_{0}$ = $_{U$_{n}$}$Γ$_{n}$ − P 2 $_{0}$Γ$_{n}$ + P 2 $_{0}$Γ$_{n}$ − ψ$_{0}$ = $^{(}$$_{U$_{n}$}$− P 2 0 $^{)}$Γ$_{n}$ + Rem Ψ n = $_{U$_{n}$}$Γ$_{0}$ + 2 ( P$_{n}$ − P$_{0}$ ) P$_{0}$ Γ$_{n}$ +$_{U}$$_{n}$ $^{(}$˜ Γ$_{n}$ − Γ$_{0}$ ) + Rem Ψ n . (4) ψ$_{n,NR}$ ≜ Ψ( ˆ P$_{n}$ ) + P$_{n}$D Ψ 1 ( ˆ P$_{n}$ ) + 1 2 $_{U$_{n}$}$D Ψ 2 ( ˆ P$_{n}$ ) , 3.2 Inference under the null 3.2.1 Asymptotic behavior ˜ Γ$_{P}$ ( o$_{1}$, o$_{2}$ ) ≜ Γ$_{P}$ ( o$_{1}$, o$_{2}$ ) − ∫ Γ$_{P}$ ( o$_{1}$, o ) dP$_{0}$ ( o ) − ∫ Γ$_{P}$ ( o, o$_{2}$ ) dP$_{0}$ ( o ) + P 2 $_{0}$Γ$_{P}$