Detecting possibly frequent change-points: Wild Binary Segmentation 2 and steepest-drop model selection-rejoinder

Many existing procedures for detecting multiple change-points in data sequences fail in frequent-change-point scenarios. This article proposes a new change-point detection methodology designed to work well in both infrequent and frequent change-point settings. It is made up of two ingredients: one is "Wild Binary Segmentation 2" (WBS2), a recursive algorithm for producing what we call a 'complete' solution path to the change-point detection problem, i.e. a sequence of estimated nested models containing $$ change-points, where T is the data length. The other ingredient is a new model selection procedure, referred to as "Steepest Drop to Low Levels" (SDLL). The SDLL criterion acts on the WBS2 solution path, and, unlike many existing model selection procedures for change-point problems, it is not penalty-based, and only uses thresholding as a certain discrete secondary check. The resulting WBS2.SDLL procedure, combining both ingredients, is shown to be consistent, and to significantly outperform the competition in the frequent change-point scenarios tested. WBS2.SDLL is fast, easy to code and does not require the choice of a window or span parameter.