the covering number N ( ϵ, µ, F , F ) as the cardinality of the smallest subcollection F ∗ ⊆ F such that, for all f ∈ F , min$_{f}$ $^{∗}$∈F ∗ ‖ f − f $^{∗}$‖$_{2}$$_{,µ}$ ≤ ϵ ‖ F ‖$_{2}$$_{,µ}$ . When H$_{0}$ holds, our proofs rely on $_{U$_{n}$}$( ˜ Γ$_{n}$ − Γ$_{0}$) = o$_{P}$$_{0}$ ( n − $^{1}$) for our method to control the type I error rate. This rate turns out to be plausible, but requires techniques which are different from the now classical empirical process techniques which can be used to establish that ( P$_{n}$ − P$_{0}$ )( f$_{n}$ − f$_{0}$ ) = o$_{P}$$_{0}$ ( n − 1 / $^{2}$) provided P$_{0}$ ( f$_{n}$ − f$_{0}$ ) 2 → 0 in probability. We ignore measurability concerns in this appendix with the understanding that minor modifications are needed to make these results rigorous. We remind the reader that a function g : O 2 →$_{R}$ is called one-degenerate if and only if g is symmetric in its arguments and P$_{0}$g ( o, · ) = 0 for all o ∈ O . Let G denote a collection of one-degenerate functions mapping from O 2 to$_{R}$, where sup$_{g}$ | g ( o$_{1}$, o$_{2}$ ) | < G ( o$_{1}$, o$_{2}$ ) for all o$_{1}$, o$_{2}$ and some envelope function G ∈ L $^{2}$( P$_{0}$ ). Suppose we wish to estimate some g$_{0}$ ∈ G . We are given a sequence of estimates ˆ g$_{n}$ ∈ G that is consistent for g$_{0}$ . Our objective is to show that The uniform entropy integral of G is given by where the supremum is over all distributions Q with support O and ‖ G ‖$_{Q,}$$_{2}$ > 0. We note that the above definition of the entropy integral upper bounds the covering integral given by Nolan and Pollard (1987), which considers a particular choice of Q . The entropy integral above lacks the square root around the logarithm in the integral that is seen in the standard definition of the uniform entropy integral used to bound empirical processes (see, e.g., van der 37 J ( t, G , G ) ≜ sup Q ∫ t 0 log N ( ϵ, Q, G , G ) dϵ, (A.1) n$_{U}$$_{n}$ (ˆ g$_{n}$ − g$_{0}$ ) = o$_{P}$$_{0}$ (1) . Appendix B.1: Bounding U -processes