content_image/319791.jpg content_image/319790.jpg content_image/319789.jpg content_image/319788.jpg
would expect 

<img src='content_image/319791.jpg'>

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 

<img src='content_image/319790.jpg'>

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 

<img src='content_image/319789.jpg'>

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 

<img src='content_image/319788.jpg'>

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 

