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Technical details

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-Figure 1: Left: One simulated dataset generated according to -\(\mathcal{MN}_{100\times -2}(\textbf{0}_{100\times 2}, \textbf{I}_{100}, \sigma^2 -\textbf{I}_{2})\). We apply \(k\)-means clustering to obtain three -clusters. The cluster centroids are displayed as triangles. -Center: Quantile-quantile plot of the Wald p-values applied to -2,000 simulated datasets from \(\mathcal{MN}_{100\times 2}(\textbf{0}_{100\times -2}, \textbf{I}_{100}, \sigma^2 \textbf{I}_{2})\). Right: -Quantile-quantile plot of our proposed p-values applied to the same -simulated datasets as in center. +
+Figure 1: We simulated one dataset according to \(\mathcal{MN}_{100\times 10}(\textbf{\mu}, +\textbf{I}_{100}, \Sigma)\), where \(\mu_i = (1,0_9)^T\) for \(i=1,\ldots, 50\) and \(\mu_i = (0_9,1)^T\) for \(i=51,\ldots, 100\), and \(\Sigma_{ij} = 1\{i=j\}+0.4\cdot 1\{i\neq +j\}\). (a): Empirical distribution of feature 2 based on +the simulated data set. In this case, all observations have the same +mean for feature 2. (b): We apply k-means clustering to obtain +two clusters and plot the empirical distribution of feature 2 stratified +by the clusters. (c): Quantile-quantile plot of naive z-test +(black) our proposed p-values (orange) applied to the simulated data +sets for testing the null hypotheses for a difference in means for +features 2–8.
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