omnibus nature of the test implies that the confidence set does not contain functions f that are “far away” from T$_{0}$ , contrary to a test which has no power against certain alternatives. Formalization of this claim is an area of future research. The authors thank Noah Simon for helpful discussions. Alex Luedtke was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. Marco Carone was supported by a Genentech Endowed Professorship at the University of Washington. Mark van der Laan was supported by NIH grant R01 AI074345-06. We now review first- and second-order pathwise differentiability. Define the following fluc- tuation submodel through P$_{0}$ : The function h$_{1}$ is a score, and the closure of the linear span of all such scores yields the unrestricted tangent space L 2 $_{0}$( P$_{0}$ ), i.e. the set of P$_{0}$ mean zero functions in L $^{2}$( P$_{0}$ ). Note that it is the resulting (first-order) tangent space that is important, as all differentiability 35 dP$_{t}$ ( o ) ≜ $^{(}$1 + th$_{1}$ ( o ) + t $^{2}$h$_{2}$ ( o ) $^{)}$dP$_{0}$ ( o ) , P$_{0}$h$_{j}$ = 0 and sup o ∈O | h$_{j}$ ( o ) | < ∞ , j = 1 , 2 . where Appendix A: Pathwise differentiability Acknowledgement 4. To improve upon our proposal for nonparametrically testing variable importance via the conditional mean function, as discussed in Section 5, it may be fruitful to consider the related Hilbert Schmidt independence criterion (Gretton et al., 2005). Higher-order pathwise differentiability may prove useful to estimate and infer about this discrepancy measure.