Suppose now that the data unit only consists of O ≜ ( W,Y ). If R$_{P}$ ( o ) ≜ E$_{P}$ ( Y | W = w ) and S$_{P}$ ≡ 0, the null hypothesis corresponds to the outcome Y having conditional mean zero in all strata of covariates. This may be interesting when zero has a special importance for the outcome, such as when the outcome is the profit over some period. Suppose again that O ≜ ( W,Y ) and W ≜ ( W (1) ,W (2) ,...,W ( K )). Denote by W ( − k ) the vector ( W ( i ) : 1 ≤ i ≤ K,i = k ). Setting R$_{P}$ ( o ) ≜ E$_{P}$ ( Y | W = w ) and S$_{P}$ ( o ) ≜ E$_{P}$ ( Y | W ( − k ) = w ( − k )), the null hypothesis corresponds to W ( k ) hav- ing null variable importance in the presence of W ( − k ) with respect to the conditional mean of Y given W in the sense that E$_{P}$ ( Y | W ) = E$_{P}$ ( Y | W ( − k )) almost surely. This is true because if R$_{0}$ ( W ) d = S$_{0}$ ( W ( − k )), the latter random variables have equal variance and so implying that V ar$_{P}$$_{0}$ [ R$_{0}$ ( W ) | W ( − k )] = 0 almost surely. Thus, a test of R$_{P}$ ( O ) d = S$_{P}$ ( O ) is equivalent to a test of almost sure equality between R$_{P}$ and S$_{P}$ in this example. We will show in Section 5 that our approach cannot be directly applied to this example, but that a simple extension yields a valid test. Gretton et al. (2006) investigated the related problem of testing equality between two distributions in a two-sample problem. They proposed estimating the maximum mean dis- crepancy (hereafter referred to as MMD), a non-negative numeric summary that equals zero if and only if the two distributions are equal. They also investigated related problems using this technique (see, e.g., Gretton et al., 2009, 2012; Sejdinovic et al., 2013). In this work, we also utilize the MMD as a parsimonious summary of equality but consider the more general 4 Example 4: Testing a null variable importance. E$_{P}$$_{0}$ { V ar$_{P}$$_{0}$ [ R$_{0}$ ( W ) | W ( − k )] } = V ar$_{P}$$_{0}$ [ R$_{0}$ ( W )] − V ar$_{P}$$_{0}$ { E$_{P}$$_{0}$ [ R$_{0}$ ( W ) | W ( − k )] } = V ar$_{P}$$_{0}$ [ R$_{0}$ ( W )] − V ar$_{P}$$_{0}$ [ S$_{0}$ ( W ( − k ))] = 0 ,