Conference Agenda

In this work, we develop sequential tests for a change in the mean of a dependent, Banach space-valued time series. For this purpose, we introduce a new class of weighted CUSUM statistics tailored to the detection of changes shortly after they occur. Unlike current alternatives -which either experience long detection delays or offer short delays only at the very beginning of the monitoring period- our approach provides consistently short detection delays anywhere in the monitoring period. This property is highly relevant for modern applications, such as epidemiology and finance, where short delays are crucial and the timing of the change is unpredictable. Our theoretical results are based on new Hölderian invariance principles that we prove under some high level conditions for Banach space-valued data. We show that these conditions hold in may instances and discuss as one particular example m-approximable time series on Hilbert spaces. Such time series cover important classes of models for multivariate and functional data. The resulting Hölderian invariance principle for m-approximable time series is of independent interest. A simulation study and data example underline the usefulness and relative advantages of the proposed approach.

A generalization of the CUSUM statistic can be used to detect structural breaks in functional time series. However, to determine the critical value without the help of dimension reduction is challenging. We propose a sequential version of the functional autoregressive sieve bootstrap and show that this bootstrap method is asymptotically valid. Additionally to the theoretical results, we demonstrate that it leads to well calibrated test for finite samples in a simulation study. We compare the AR-Sieve Bootstrap to other resampling methods suggested: block bootstrap and dependent wild bootstrap.

Change point detection methods are a common tool to identify structural changes in the distribution of time series. In recent years, there has been progress in detecting changes within times series in countable spaces, e.g. the natural numbers. For a number of applications, such as outbreak detection of infectious diseases, modeling a gradual change could be valuable. Such count time series can be modeled by Poisson INARCH(1) processes. One possibility to model gradual changes is by introducing a non-linear time dependent factor in the intensity function of a Poisson INARCH(1) process. This additional factor characterizes the gradual change after the change point.Two test statistics are applied to this model and it is shown that both still have a limiting distribution given by the Gumbel extreme value distribution under the null hypothesis. Under the alternative, consistency holds for certain assumptions. The difference between both test statistics is elaborated by an experimental analysis.