definition of D R ˆ $_{P$_{n}$}$( o ) may be significant. Thus, without some form of sample splitting, the finite sample performance of ψ$_{n}$ may be poor even under the conditions stated in Appendix B.1. As indicated above, our test consists of rejecting H$_{0}$ if and only if ψ$_{n}$ is larger than some cutoff c$_{n}$ . We wish to select c$_{n}$ to yield a non-conservative test at level α ∈ (0 , 1). In view of Theorem 4, denoting by q$_{1}$$_{−}$$_{α}$ the 1 − α quantile of the described limit distribution, the cutoff c$_{n}$ should be chosen to be q$_{1}$$_{−}$$_{α}$/n . We thus reject H$_{0}$ if and only if nψ$_{n}$ > q$_{1}$$_{−}$$_{α}$ . As described in the following corollary, q$_{1}$$_{−}$$_{α}$ admits a very simple form when S$_{P}$ ≡ 0 for all P . Corollary 2 (Asymptotic distribution under H$_{0}$ , S degenerate) . Suppose that C1), C2) and C3) hold, that S$_{P}$ ≡ 0 for all P ∈ M , and that σ 2 R ≜ V ar$_{P}$$_{0}$ $^{[}$D R 0 ( O ) ] > 0 . Then, under H$_{0}$ , where Z is a standard normal random variable. It follows then that q$_{1}$$_{−}$$_{α}$ = 2 σ 2 $_{R}$( z 2 1 − α/ 2 − 1) , where z$_{1}$$_{−}$$_{α/}$$_{2}$ is the (1 − α/ 2) quantile of the standard normal distribution. The above corollary gives an expression for q$_{1}$$_{−}$$_{α}$ that can easily be consistently estimated from the data. In particular, one can use ˆ q$_{1}$$_{−}$$_{α}$ ≜ 2( z 2 1 − α/ 2 − 1) P$_{n}$D $^{R}$( ˆ P$_{n}$ ) 2 as an estimator of q$_{1}$$_{−}$$_{α}$ , whose consistency can be established under a Glivenko-Cantelli and consistency condition on the estimator of D R 0 . However, in general, such a simple expression will not exist. Gretton et al. (2009) proposed estimating the eigenvalues ν$_{k}$ of the centered Gram matrix and then computing ˆ λ$_{k}$ ≜ ν$_{k}$/n . In our context, the eigenvalues ν$_{k}$ are those of the n × n matrix G ≜ { G$_{ij}$ }$_{1}$$_{≤}$$_{i,j}$$_{≤}$$_{n}$ with entries defined as 17 G$_{ij}$ ≜ Γ$_{n}$( O$_{i}$, O$_{j}$ ) − 1 n n ∑ k =1 Γ$_{n}$( O$_{k}$, O$_{j}$ ) − 1 n n ∑ ℓ =1 Γ$_{n}$( O$_{i}$, O$_{ℓ}$ ) + 1 n 2 n ∑ k =1 n ∑ ℓ =1 Γ$_{n}$( O$_{k}$, O$_{ℓ}$ ) . (5) nψ$_{n}$ 2 σ 2 R ⇝ Z 2 − 1 , 3.2.2 Estimation of the test cutoff