Binomial, with p = 0 . 119 so that the penalty becomes that of the Akaike information criterion (i.e., log π ( I| m ) π ( m ) = − 2 m ); π ( m ) = 1 /K for fixed K for Uniform. We also considered the prior in Jiang et al. (2014), resulting in the DS test with penalty − λ$_{0}$ log N ( m − 1). According to the recommendation in Jiang et al. (2014), we chose the value of λ$_{0}$ so that the level under the null is as close as possible to 0.05 from below, so λ$_{0}$ = 1 . 11088 for N = 100, and λ$_{0}$ = 0 . 904 for N = 500. The Poisson prior (with rate √ N ) was by far the best among all priors considered, and its power was comparable to that of the minimum p -value displayed in Table 6. For sample size N = 100, we examined the distributions depicted in Figure 10, and we used 20000 simulated data sets, in each of the configurations. Table 9 and Figure 11 show the power for the setups in Figure 10. These results concur with the results in Section 4.1 for N = 500, and show that if the number of intersections of the two distributions is at least four, tests statistics with m ≥ 4 have an advantage. 67 Table 7: Power using different priors , for N = 500 and the setups of Figure 2. Table 8: Power of competitors (columns 4–9), along with the minimum p -value using the M$_{m}$ p -values (column 1) and the S$_{m}$ p -values (column 2). Poisson prior Poisson prior Uniform Prior Binomial Prior DS prior Setup Max aggregation Sum aggregation Max aggregation Max aggregation Max aggregation 1 0.829 0.597 0.848 0.450 0.845 2 0.855 0.924 0.641 0.745 0.565 3 0.715 0.639 0.282 0.874 0.239 4 0.861 0.920 0.813 0.593 0.794 5 0.659 0.746 0.592 0.411 0.573 6 0.854 0.874 0.804 0.597 0.787 7 0.291 0.310 0.430 0.139 0.432 8 0.719 0.760 0.835 0.371 0.836 9 0.489 0.545 0.599 0.232 0.600 10 0.709 0.793 0.749 0.406 0.744 11 0.607 0.720 0.466 0.390 0.433 12 0.460 0.629 0.298 0.318 0.268 13 0.860 0.857 0.802 0.697 0.776 Minimum p -value N Setup Max aggregation Sum Aggregation Wilcoxon KS CVM AD HHG DS 500 Normal shift 0.59 0.82 0.91 0.81 0.89 0.90 0.85 0.73 500 Normal scale 0.76 0.85 0.05 0.36 0.46 0.77 0.88 0.83 500 Normal shift & scale 0.83 0.90 0.46 0.69 0.74 0.88 0.92 0.90 100 Normal shift 0.39 0.58 0.68 0.53 0.65 0.67 0.60 0.49 100 Normal scale 0.51 0.59 0.06 0.20 0.23 0.41 0.63 0.54 100 Normal shift & scale 0.54 0.64 0.49 0.51 0.58 0.64 0.68 0.62