The surrogate band and its three readings

Abstract

The multifractality test this literature has run since 2010 compares a series' singularity-spectrum width to the widths of its IAAFT surrogates and rejects when the original lies outside "the 95% confidence interval" of the surrogate widths. That phrase is not one test. On one and the same ensemble it admits at least three: a band (the surrogate mean plus 1.96 SD of the surrogate widths), a standard error of the surrogate mean (the mean plus t × SD/√n_surr — the construction the successor instrument t_MF standardised), and a rank or exchangeability test (the original must out-rank its surrogates at p < 0.05). The readings differ by a factor of √n_surr — about 5.5 at the 30 surrogates the 2010 precedent fixed — and the difference is not cosmetic. On 987 public lexical-decision session series, the reconstructed 2010 rule flags 51/987 under the band reading and 63/987 under the rank reading — the nominal rate — while the same thirty surrogates read through the surrogate mean's standard error flag 478/987, and the successor instrument's published rule flags 495/987 significant-positive plus 252 significant-negative. On identical ensembles at the same length, the standard-error reading fires significant-positive on 8/24 realisations of a fitted fGn, 8/24 of a fitted AR(4), 8/24 of a Gaussian monofractal carrying the cohort's own response-time marginal and 10/24 of one carrying a single lapse episode, where the band readings of the same ensembles fire 0–2/24 and the rank readings 0–1/24 — at full power on the canonical cascade. The same behaviour replicates on a different lineage member's own published recording. The convention, not the statistic, is the cost: the sealed flag says so, and the fix is one line of a methods section. Report the band, the standard error and the rank side by side, both tails, and state n_surr.