Comparison Of Methods For Testing Stationarity Of Functional Time Series Against Change Point Alternative

Comparing with traditional (scalar or vector) time series, research on functional time series started late, so it’s still steadily growing and has not formed a complete theoretical system. Testing stationarity is an initial and critical issue when doing time series analysis, and testing stationarity against change point alternative is a fundamen-tal direction for research.Considering that the assumption of independence is rather strong and could not fit well in many practical cases, thus we focus on a relatively weaker condition:Lk-m-approximability weak dependence. We first give a summary of three different types of test statistics introduced in Horvath et al (2014),Torgovitski (2014b) and derive the limit distributions under the null-hypothesis. Then we demonstrate consistency of tests in accordance with asymptotic behavior of statistics under abrupt change point alternative.In this paper, we extend the results, from the alternative hypothesis of abrupt one-change point case to gradual multi-change points case. We study asymptotic properties of three types of statistics and detailed proofs are presented. It shows that three types of test methods are still consistent against gradual change case. In the experimental studies, we conduct a Monte Carlo simulation. We consider both i.i.d Brownian motion and FAR(1) model in data generating processes, implement these three types of test methods simultaneously and draw some conclusions by comparing the empirical sizes and power.