each of R$_{0}$ and S$_{0}$ . The approach we propose will apply to functionals within a specified class described later. Before presenting our general approach, we describe some motivating examples. Consider the data structure O = ( W,A,Y ) ∼ P , where W is a collection of covariates, A is binary treatment indicator, and Y is a bounded outcome, i.e., there exists a universal c such that, for all P ∈ M , P ( | Y | ≤ c ) = 1. Note that, in our examples, the condition that Y is bounded cannot easily be relaxed, as the parameter from Gretton et al. (2006) on which we will base our testing procedure requires that the quantities under consideration have compact support. If R$_{P}$ ( o ) ≜ E$_{P}$ ( Y | A = 1 ,W = w ) − E$_{P}$ ( Y | A = 0 ,W = w ) and S$_{P}$ ≡ 0, the null hypothesis corresponds to the absence of a conditional average treatment effect. This definition of R$_{P}$ corresponds to the so-called blip function introduced by Robins (2004), which plays a critical role in defining optimal personalized treatment strategies (Chakraborty and Moodie, 2013). Suppose the setting of the previous example, but where A represents membership to population 0 or 1. If R$_{P}$ ( o ) ≜ E$_{P}$ ( Y | A = 1 ,W = w ) and S$_{P}$ ( o ) ≜ E$_{P}$ ( Y | A = 0 ,W = w ), the null hypothesis corresponds to the outcome having the conditional mean functions, applied to a random draw of the covariate, having the same distribution in these two populations. We note here that our formulation considers selection of individuals from either population as random rather than fixed so that population-specific sample sizes (as opposed to the total sample size) are themselves random. The same interpretation could also be used for the previous example, now testing if the two regression functions are equivalent. 3 Example 3: Testing a null covariate effect on average response. Example 2: Testing for equality in distribution of regression functions in two popula- tions. Example 1: Testing a null conditional average treatment effect.