Table 1: Power of competitors (columns 4-9), along with the minimum p -value statis- tic using the M$_{m}$ p -values (column 2) and the S$_{m}$ p -values (column 3), for N = 500. The score per partition was the likelihood ratio test statitsic. The standard error was at most 0.0035. The advantage of the test based on the minimum p -value is large when the number of intersections of the two densities is at least four (setups 2,3,4,5,6,10,11,12, and 13). The best competitors are HHG and DS, but HHG is es- sentially an m ≤ 3 test, and DS penalizes large m s severely, therefore in setups where m ≥ 4 partitions are better they can perform poorly. Among the two variants in columns 2 and 3, the better choice clearly depends on the range of support in which the differences in distributions occur: aggregation by maximum has better power when the difference between the distributions is very local (setups 1, 3, and 13), and aggregation by summation has better power otherwise. The highest power per row is underlined. Min p -value aggreg. Setup by Max by Sum Wilcoxon KS CVM AD HHG DS 1 Normal vs. Normal with delta 0.825 0.491 0.072 0.149 0.108 0.099 0.175 0.849 2 Mix. Vs. Mix., 3 Vs. 4 Components 0.799 0.873 0.000 0.020 0.001 0.021 0.344 0.560 3 Normal vs. Normal with many deltas 0.785 0.733 0.051 0.078 0.073 0.099 0.142 0.245 4 Normal vs. Mixture 2 Components 0.827 0.937 0.053 0.531 0.458 0.495 0.855 0.796 5 Normal vs. Mixture 5 Components 0.592 0.686 0.048 0.238 0.179 0.246 0.484 0.556 6 Normal vs. Mixture 10 Components 0.818 0.820 0.048 0.240 0.211 0.310 0.561 0.789 7 Cauchy, Shift 0.339 0.492 0.542 0.620 0.627 0.577 0.641 0.436 8 Symmetric Gaussian mixture 0.752 0.775 0.033 0.194 0.242 0.617 0.749 0.835 9 Asymmetric Gaussian mixture 0.512 0.613 0.050 0.253 0.277 0.469 0.678 0.599 10 Asymmetric Mixture vs. Mixture 0.711 0.806 0.000 0.159 0.119 0.395 0.690 0.747 11 Mix. Vs. Mix., 2 Vs. 3 Components 0.540 0.686 0.004 0.093 0.057 0.116 0.302 0.440 12 Mix. Vs. Mix., 2 Vs. 4 Components, Symmetric 0.390 0.577 0.000 0.005 0.000 0.005 0.079 0.270 13 Mix. Vs. Mix., 3 Vs. 3 Components, Asymmetric 0.844 0.764 0.000 0.001 0.000 0.013 0.042 0.780 14 Null 0.051 0.051 0.050 0.043 0.050 0.050 0.050 0.050 first transform the observations of each variable into ranks, referred to as dCov and HHG, respectively. We note that the power of the original dCov and HHG was fairly similar to the power of their distribution-free variants, see Appendix I. We examine complex bivariate relationships depicted in Figure 4. Most of these sce- narios were collected from the literature illustrating the performance of other meth- ods. Specifically, the first two rows were examined in Newton (2009), the next two rows are similar to the relationships examined in Reshef et al. (2011), and the Heav- isine and Doppler examples in the last row were used extensively in the literature on denoising, see e.g., Donoho and Johnstone (1995). In all but the 4 Independent Clouds setup, there is dependence. The 4 Independent Clouds setup allows us to 32