content_image/308046.jpg content_image/308047.jpg content_image/308042.jpg
from the nonparametric model of distributions P ′ satisfying the same constraints as P to the real line. This functional is pathwise differentiable at P relative to the maximal tangent space with an efficient influence function f$_{π}$ given by 

<img src='content_image/308046.jpg'>

By the bound on Y and the strong positivity assumption, there exists a universal constant M > 0 such that P (sup$_{π}$$_{∈}$$_{Π}$ | f$_{π}$ ( O ) | ≤ M ) = 1. 

We refer the interested reader to [14, Section 25.3] and [7] for definitions and proofs of these facts. Although the class 

<img src='content_image/308047.jpg'>

will play a central role in the rest of this article, it is not necessary to master the derivation or properties of efficient influence functions to appreciate the contributions of this work. 

## 2 Main Results 

## 2.1 Preliminary 

Suppose that we observe mutually independent data-structures O$_{1}$ ≡ ( X$_{1}$, A$_{1}$, Y$_{1}$ ) , . . . , O$_{n}$ ≡ ( X$_{n}$, A$_{n}$, Y$_{n}$ ) drawn from P . Let P$_{n}$ ≡ n − 1 ∑ n i $_{=1}$Dirac( O$_{i}$ ) denote the empirical measure. For every function f mapping an observation to the real line, we set Pf ≡ E [ f ( X, A, Y )] and P$_{n}$f ≡ n − 1 ∑ n i $_{=1}$f ( O$_{i}$ ). Our theorem will attain fast rates under the assumption that one has available an esti- 

mator { ̂ V ( π ) : π ∈ Π } of {V ( π ) : π ∈ Π } that makes the following term small: 

<img src='content_image/308042.jpg'>

