Robust Narrowest Significance Pursuit: Inference for Multiple Change-Points in the Median

We propose Robust Narrowest Significance Pursuit (RNSP), a methodology for detecting localized regions in data sequences which each must contain a change-point in the median, at a prescribed global significance level. RNSP works by fitting the postulated constant model over many regions of the data using a new sign-multiresolution sup-norm-type loss, and greedily identifying the shortest intervals on which the constancy is significantly violated. By working with the signs of the data around fitted model candidates, RNSP fulfils its coverage promises under minimal assumptions, requiring only sign-symmetry and serial independence of the signs of the true residuals. In particular, it permits their heterogeneity and arbitrarily heavy tails. The intervals of significance returned by RNSP have a finite-sample character, are unconditional in nature and do not rely on any assumptions on the true signal. Code implementing RNSP is available at https://github.com/pfryz/nsp.