3 Proposed test: formulation and inference under the null 3.1 Formulation of test We begin by constructing an estimator of ψ$_{0}$ from which a test can then be devised. Using the fact that Ψ( P ) = P $^{2}$Γ$_{P}$ , as implied by (3), we note that if Γ$_{0}$ were known, the U-statistic $_{U$_{n}$}$Γ$_{0}$ would be a natural estimator of ψ$_{0}$ , where U$_{n}$ denotes the empirical measure that places equal probability mass on each of the n ( n − 1) points ( O$_{i}$, O$_{j}$ ) with i = j . In practice, Γ$_{0}$ is unknown and must be estimated. This leads to the estimator ψ$_{n}$ ≜ $_{U$_{n}$}$Γ$_{n}$ , where we write Γ$_{n}$ ≜ Γ$_{ˆ}$ P$_{n}$ for some estimator ˆ P$_{n}$ of P$_{0}$ based on the available data. Since a large value of ψ$_{n}$ is inconsistent with H$_{0}$ , we will reject H$_{0}$ if and only if ψ$_{n}$ > c$_{n}$ for some appropriately chosen cutoff c$_{n}$ . In the nonparametric model considered, it may be necessary, or at the very least desirable, to utilize a data-adaptive estimator ˆ P$_{n}$ of P$_{0}$ when constructing Γ$_{n}$. Studying the large- sample properties of ψ$_{n}$ may then seem particularly daunting since at first glance we may be led to believe that the behavior of ψ$_{n}$ − ψ$_{0}$ is dominated by P 2 0 (Γ$_{n}$ − Γ$_{0}$). However, this is not the case. As we will see, under some conditions, ψ$_{n}$ − ψ$_{0}$ will approximately behave like ($_{U}$$_{n}$ − P 2 $_{0}$) Γ$_{0}$ . Thus, there will be no contribution of ˆ P$_{n}$ to the asymptotic behavior of ψ$_{n}$ − ψ$_{0}$ . Though this result may seem counterintuitive, it arises because Ψ( P ) can be expressed as P $^{2}$Γ$_{P}$ with Γ$_{P}$ a second-order gradient (or rather an extension thereof) up to a proportionality constant. More concretely, this surprising finding is a direct consequence of (3). As further support that ψ$_{n}$ is a natural test statistic, even when a data-adaptive estimator ˆ P$_{n}$ of P$_{0}$ has been used, we note that ψ$_{n}$ could also have been derived using a second-order one-step Newton-Raphson construction, as described in Robins et al. (2008). The latter is 14