How much of one search the other has already found

Five pairs of searches on one sample, on a scale whose zero and one are both fixed by construction. Zero is two searches over disjoint sets of independent columns: they remove shares of the residual sum that add, at 0.8 standard errors from exactly additive, and they read 0.004. One is a break search paired with a step column it contains, which reads exactly one on every draw because the step adds nothing at all. Between them: two dictionaries of step columns cut a few rows apart read 0.125, and the pair the earlier field measured — a break and a whitening window, both reading the same residual series — reads 0.762, three quarters of the way to one search containing the other. And below zero, a break paired with a search over independent columns reads -0.306: the joint search finds configurations neither half of it contains, so charging the two separately under-charges.

The control at four dictionary sizes, over 600 draws apiece: the two shares added, minus the share the joint search removes, with two standard errors either side. Every reading is inside three standard errors of zero — -0.85, 2.40, 0.58, 1.68 — while what each search finds grows from 0.0142 of the residual sum at 2 columns to 0.0328 at 10. So the zero is not the zero of two searches with nothing to find. It is the zero of two searches whose findings occupy directions that do not overlap, which is what a scale's origin has to mean if the numbers above it are to be read as shares of one search that the other has already taken.

Each pair's overlap on the doubled log-likelihood scale — the sum of the two searches' own quantiles less the joint search's quantile, as a share of the smaller — at five quantile levels, over 1200 draws under the first-order autoregression. two dictionaries of independent columns: -0.176 at 0.5, -0.000 at 0.75, 0.238 at 0.9, 0.320 at 0.95, 0.538 at 0.99; a break, and an independent column: -1.211 at 0.5, -0.406 at 0.75, -0.029 at 0.9, 0.205 at 0.95, 0.161 at 0.99; two dictionaries of step columns: -0.397 at 0.5, 0.145 at 0.75, 0.438 at 0.9, 0.541 at 0.95, 0.682 at 0.99; a break, and a step column it contains: 1.000 at 0.5, 1.000 at 0.75, 1.000 at 0.9, 1.000 at 0.95, 1.000 at 0.99; a break, and a whitening window: 0.512 at 0.5, 0.640 at 0.75, 0.716 at 0.9, 0.732 at 0.95, 0.754 at 0.99. Every pair but the containment overlaps more at the 95% point than at the median, the control included.

How often each of four rules says something is there, for the pair of searches over a break and a column, under four truths, 400 draws each under a first-order autoregression at 0.8 with every threshold simulated on 1200 null draws. nothing there: joint, calibrated 6.3%, summed thresholds 6.3%, Bonferroni 6.3%, either, uncorrected 12.0%. a shift from row 60: joint, calibrated 65.3%, summed thresholds 64.0%, Bonferroni 55.0%, either, uncorrected 69.3%. a column enters: joint, calibrated 11.8%, summed thresholds 10.8%, Bonferroni 73.0%, either, uncorrected 82.5%. both: joint, calibrated 70.5%, summed thresholds 69.0%, Bonferroni 75.8%, either, uncorrected 84.8%. The first block is each rule's size.