Asymptotic fluctuations of mutagrams

Abrupt changes in sequential data, known as change-points, can be detected by fitting differences of one-sided kernel estimators. The detection depends on both a bandwidth and a threshold, where we declare a change-point whenever the differences of the one-sided kernel estimators exceed the threshold. The joint behavior of the difference of one-sided kernel estimators and scale can be graphically represented in a mutagram [Müller, H.G., Wai, N., 2004. Change trees and mutagrams for the visualization of local changes in sequence data. J. Comput. Statist. Graph. 13, 571-585.]. Here we study the fluctuations of the mutagram values in dependency on different bandwidth choices, corresponding to scale. Study of these fluctuations is related to the SiZer approach of Chaudhuri and Marron [2000. Scale space view of curve estimation. Ann. Statist. 28, 408-428.]. We explore the asymptotic properties and convergence of a local bandwidth process for differences of one-sided kernel estimators. These processes are shown to be tight with a Gaussian limit process that represents the asymptotic fluctuations of mutagrams. These fluctuations are illustrated with a sequence of vertical ocean shear data.