0.05 0.10 0.15 0.20 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 # Signal: 0 5 10 15 20 125 250 500 1000 2000 Sample Size Probability of Rejecting the Null Bandwidth 1 1/2 Figure 4: Probability of rejecting the null when testing the null hypothesis that µ (1 ,W ) − µ (0 ,W ) is almost surely equal to zero in Simulation 3. signal coordinate j , given A and W , 20 Y$_{j}$ was drawn from the same conditional distribution as Y give A and W in Simulation 1c. For noise coordinate j , given A and W , 20 Y$_{j}$ was drawn from the same conditional distribution as Y given A and W in Simulation 1a. Relative to Simulation 1, we have scaled each coordinate of the outcome to be one twentieth the size of the outcome in Simulation 1. Apart from the Gaussian kernel with bandwidth one, which we have adopted throughout this paper, we considered defining the MMD with a Gaussian kernel with bandwidth 1 / 2. Alternatively, this could be viewed as considering bandwidths 1 / 20 and 1 / 40 if the outcome had not been scaled by 1 / 20. We ran the same Super Learner to estimate µ (1 ,w ) as in Simulation 1, and we again treated the probability of treatment given covariates as known. We evaluated significance by estimating all of the positive eigenvalues of the centered Gram matrix for n = 125 and the largest 200 positive eigenvalues of the centered Gram matrix for n > 125. In Figure 4, the empirical null rejection probability is displayed for our proposed MMD method based on bandwidths 1 and 1/2. Our proposal appears to control type I error well 31