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literature to tackle the problem at hand. To the best of our knowledge, this is the first instance in which these techniques are directly used (without any form of approximation) to resolve an open methodological problem. 

This paper is organized as follows. In Section 2, we formally present our parameter of interest, the squared MMD between two unknown functions, and establish asymptotic representations for this parameter based on its higher-order differentiability, which, as we formally establish, holds even when the MMD involves estimation of unknown nuisance parameters. In Section 3, we discuss estimation of this parameter, discuss the corresponding hypothesis test and study its asymptotic behavior under the null. We study the consistency of our proposed test under fixed and local alternatives in Section 4. We revisit our examples in Section 5 and provide an additional example in which we can still make progress using our techniques even though our regularity conditions fails. In Section 6, we present results from a simulation study to illustrate the finite-sample performance of our test, and we end with concluding remark in Section 7. 

Appendix A reviews higher-order pathwise differentiability. Appendix B gives a summary of the empirical U -process results from Nolan and Pollard (1988) that we build upon. All proofs can be found in Appendix C. 

## 2 Properties of maximum mean discrepancy 

## 2.1 Definition 

For a distribution P and mappings T and U , we define 

<img src='content_image/315510.jpg'>

and set Ψ( P ) ≜ Φ $^{RR}$( P ) − 2Φ $^{RS}$( P ) + Φ $^{SS}$( P ). The MMD between the distributions of R$_{P}$ ( O ) and S$_{P}$ ( O ) when O ∼ P is given by $^{√}$Ψ( P ) and is always well-defined because 

