Asymptotic of the number of false change points of the fused lasso signal approximator

It is well-known that the fused lasso signal approximator (FLSA) is inconsistent in change point detection under the presence of staircase blocks in true mean values. The existing studies focus on modifying the FLSA model to remedy this inconsistency. However, the inconsistency of the FLSA does not severely degrade the performance in change point detection if the FLSA can identify all true change points and the estimated change points set is sufficiently close to the true change points set. In this study, we investigate some asymptotic properties of the FLSA under the assumption of the noise level \(\sigma _n = o(n \log n)\). To be specific, we show that all the falsely segmented blocks are sub-blocks of true staircase blocks if the noise level is sufficiently low and a tuning parameter is chosen appropriately. In addition, each false change point of the optimal FLSA estimate can be associated with a vertex of a concave majorant or a convex minorant of a discrete Brownian bridge. Based on these results, we derive an asymptotic distribution of the number of false change points and provide numerical examples supporting the theoretical results.