investigated. It is not difficult to show then that, for any T, U ∈ S$_{d}$ ( P$_{0}$ ), Γ TU $_{d,P}$( o$_{1}$, o$_{2}$ ) is given by where Id denotes the d -dimensional identity matrix and A ′ denotes the transpose of a given vector A . Using these objects, the method and results presented in this paper can be replicated in higher dimensions rather easily. 34 3. Our results can be used to develop confidence sets for infinite dimensional parameters by test inversion. Consider a parameter T satisfying our conditions. Then one can test if R$_{0}$ ≜ T$_{0}$ − f is equal in distribution to zero for any fixed function f that does not rely on P . Under the conditions given in this paper, a 1 − α confidence set for T$_{0}$ is given by all functions f for which we do not reject H$_{0}$ at level α . The blip function from Example 1 is a particularly interesting example, since a confidence set for this parameter can be mapped into a confidence set for the sign of the blip function, i.e. the optimal individualized treatment strategy (Robins, 2004). We would hope that the Φ TU d ( P ) ≜ ∫ ∫ e −‖ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ ) ‖ $^{2}$dP ( o$_{1}$ ) dP ( o$_{2}$ ) . 2. While our paper focuses on univariate hypotheses, our results can be generalized to higher dimensions. Suppose that P ↦→ R$_{P}$ and P ↦→ S$_{P}$ are$_{R}$ $^{d}$-valued functions on O . The class S$_{d}$ of allowed such parameters can be defined similarly as S , with all original conditions applying componentwise. The MMD for the vector-valued parameters R and S using the Gaussian kernel is given by Ψ$_{d}$( P ) ≜ Φ RR d ( P ) − 2Φ RS d ( P ) + Φ SS d ( P ), where for any T, U ∈ S$_{d}$ we set $^{[}$2 [ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ )] ′ $^{[}$D U $_{P}$( o$_{2}$ ) − D T $_{P}$( o$_{1}$ ) ] + 1 − 2 D T $_{P}$( o$_{1}$ ) ′ $^{{}$2 [ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ )] [ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ )] ′ − Id $^{}}$D U $_{P}$( o$_{2}$ ) ] × e −‖ T$_{P}$ ( o$_{1}$ ) − U$_{P}$ ( o$_{2}$ ) ‖ $^{2}$,