Combining Cumulative Sum Change‐Point Detection Tests for Assessing the Stationarity of Univariate Time Series

Abstract

We derive tests of stationarity for univariate time series by combining change‐point tests sensitive to changes in the contemporary distribution with tests sensitive to changes in the serial dependence. The proposed approach relies on a general procedure for combining dependent tests based on resampling. After proving the asymptotic validity of the combining procedure under the conjunction of null hypotheses and investigating its consistency, we study rank‐based tests of stationarity by combining cumulative sum change‐point tests based on the contemporary empirical distribution function and on the empirical autocopula at a given lag. Extensions based on tests solely focusing on second‐order characteristics are proposed next. The finite‐sample behaviors of all the derived statistical procedures for assessing stationarity are investigated in large‐scale Monte Carlo experiments, and illustrations on two real datasets are provided. Extensions to multi‐variate time series are briefly discussed as well.

Nonparametric change point detection for periodic time series

“…The R package npcp provides nonparametric CUSUM tests for detecting changes in possibly serially dependent univariate or multivariate observations. Kojadinovic (2017) implemented this routine based on the works from Holmes, Kojadinovic & Quessy (2013), Bücher & Kojadinovic (2016) and Bücher, Fermanian & Kojadinovic (2017). We use the function cpDist with statistics option ksmax in the package, and we refer to this method as Empirical.…”

Detecting deviations from second-order stationarity in locally stationary functional time series

“…Consistency of this procedure is a mere consequence of Proposition 2.1 in Bücher et al (2018); details are omitted for the sake of brevity.…”

“…As explained in the next paragraph, the individual tests are then combined to yield a joint test including autocovariances up to a given maximal lag H. Thus, the approach investigated here is similar in spirit to the classical Portmanteau tests for serial correlation of a univariate time series, where the hypothesis of white noise is checked by investigating whether correlations up to a given lag vanish (see Box and Pierce, 1970;Ljung and Box, 1978). For the problem of checking stationarity in real-valued time series, similar approaches have been taken by Jin et al (2015) and Bücher et al (2018).…”

A practical adaptive nonlinear tracking algorithm with range rate measurement

“…28 The CUSUM change-point detection technology is applied to the stationary evaluation method of univariate time series. 29 A nonparameter CUSUM is designed for quickest detection of parameter changes in stochastic regression. 30 The CUSUM is more appropriate in the target tracking problem discussed in the article, because the measurement vector of radar comes according to the time sequence.…”