A Proof of Lemma 1 According to the second condition in (8), we have q (x) = q (x

Therefore, it fails to control the false positive rate. Figure 10: Distribution of naive p -value when the null hypothesis is true. Figure 11: Distribution of selective p -value when the null hypothesis is true. Figure 12: Uniform QQ-plot of the pivot. In the above example, we used 3 cuts (pieces) to approximate the function. Figure 13, we show that # encountered intervals still linearly increase in practice. Figure 13: Demonstration of # encountered and # truncation intervals when increasing # cuts (pieces).