would expect to be a third-order remainder term. However, second-order pathwise differentiability has only been established under the null, and in fact, it appears that Ψ may not generally be second-order pathwise differentiable under the alternative. As such, D Ψ 2 may not even be defined under the alternative. In writing (2), we either naively set D Ψ 2 ( P ) ≜ 2Γ$_{P}$ , which is not appropriately centered to be a candidate second-order gradient, or instead take D Ψ 2 to be the centered extension Both of these choices yield the same expression above because the product measure ( P − P$_{0}$ ) 2 is self-centering. The need for an extension renders it a priori unclear whether as P tends to P$_{0}$ the behavior of Rem Ψ P is similar to what is expected under more global second-order pathwise differentiability. Using the fact that Ψ( P ) = P $^{2}$Γ$_{P}$ , we can simplify the expression in (2) to As we discuss below, this remainder term can be bounded in a useful manner, which allows us to determine that it is indeed third-order. For all T ∈ S , P ∈ M and o ∈ O , we define as the remainder from the linearization of T based on the conditional gradient D T $_{P}$. Typically, Rem T $_{P}$( o ) is a second-order term. Further consideration of this term in the context of our 12 Rem T $_{P}$( o ) ≜ T$_{P}$ ( o ) − T$_{0}$ ( o ) + ∫ D T $_{P}$( o$_{1}$ ) $^{[}$dP ( o$_{1}$ | x $^{T}$) − dP$_{0}$ ( o$_{1}$ | x $^{T}$) ] Rem Ψ P = P 2 $_{0}$Γ$_{P}$ − ψ$_{0}$ . (3) ( o$_{1}$, o$_{2}$ ) ↦→ 2$^{[}$Γ$_{P}$ ( o$_{1}$, o$_{2}$ ) − ∫ Γ$_{P}$ ( o$_{1}$, o ) dP ( o ) − ∫ Γ$_{P}$ ( o, o$_{2}$ ) dP ( o ) + P $^{2}$Γ$_{P}$$^{]}$ . Rem Ψ P ≜ Ψ( P ) − Ψ( P$_{0}$ ) − ( P − P$_{0}$ ) D Ψ 1 ( P ) + 1 2 ( P − P$_{0}$ ) $^{2}$DΨ 2 ( P ) (2)