Ψ( P ) is non-negative. Indeed, denoting by ψ$_{0}$ the true parameter value Ψ( P$_{0}$ ), Theorem 3 of Gretton et al. (2006) establishes that ψ$_{0}$ equals zero if H$_{0}$ holds and is otherwise strictly positive. Though the study in Gretton et al. (2006) is restricted to two-sample problems, their proof of this result is only based upon properties of Ψ and therefore holds regardless of the sample collected. Their proof relies on the fact that two random variables X and Y with compact support are equal in distribution if and only if E [ f ( Y )] = E [ f ( X )] for every continuous function f , and uses techniques from the theory of Reproducing Kernel Hilbert Spaces (see, e.g., Berlinet and Thomas-Agnan, 2011, for a general exposition). We invite interested readers to consult Gretton et al. (2006) – and, in particular, Theorem 3 therein – for additional details. The definition of the MMD we utilize is based on the univariate Gaussian kernel with unit bandwidth, which is appropriate in view of Steinwart (2002). The results we present in this paper can be generalized to the MMD based on a Gaussian kernel of arbitrary bandwidth by simply rescaling the mappings R and S . To develop a test of H$_{0}$ , we will first construct an estimator ψ$_{n}$ of ψ$_{0}$ . In order to avoid restrictive model assumptions, we wish to use flexible estimation techniques in estimating P$_{0}$ and therefore ψ$_{0}$ . To control the operating characteristics of our test, it will be crucial to understand how to generate a parametric-rate estimator of ψ$_{0}$ . For this purpose, it is informative to first investigate the pathwise differentiability of Ψ as a parameter from M to $_{R}$. So far, we have not specified restrictions on the mappings P ↦→ R$_{P}$ and P ↦→ S$_{P}$ . However, in our developments, we will require these mappings to satisfy certain regular- ity conditions. Specifically, we will restrict our attention to elements of the class S of all mappings T for which there exists some measurable function X T defined on O , e.g. X $^{T}$( o ) = X $^{T}$( w, a, y ) = w , such that (S1) T$_{P}$ is a measurable mapping with domain { X $^{T}$( o ) : o ∈ O } and range contained in 7 2.2 First-order differentiability