Abstract
In the common time series model Xi,n=μ(i∕n)+εi,n with non-stationary errors we consider the problem of detecting a significant deviation of the mean function μ from a benchmark g(μ) (such as the initial value μ(0) or the average trend ∫01μ(t)dt). The problem is motivated by a more realistic modelling of change point analysis, where one is interested in identifying relevant deviations in a smoothly varying sequence of means (μ(i∕n))i=1,…,n and cannot assume that the sequence is piecewise constant. A test for this type of hypotheses is developed using an appropriate estimator for the integrated squared deviation of the mean function and the threshold. By a new concept of self-normalization adapted to non-stationary processes an asymptotically pivotal test for the hypothesis of a relevant deviation is constructed. The results are illustrated by means of a simulation study and a data example.
Highlights
Within the last decades, the detection of structural breaks in time series has become a very active area of research with many applications in fields like climatology, economics, engineering, genomics, hydrology, etc
A large part of the literature considers the problem of detecting changes in a piecewise constant mean function μ : [0, 1] → R, where early references assume the existence of at most one change point
The errorsi=1,...,n in model (1.1) are usually assumed to form at least a stationary process and many theoretical results for detecting multiple change points are only available for independent identically distributed error processes. These assumptions simplify the statistical analysis of structural breaks substantially, as - after removing the piecewise constant trend - one can work under the assumption of a stationary or an independent identically distributed error process and smoothing is not necessary to estimate the trend function
Summary
The detection of structural breaks in time series has become a very active area of research with many applications in fields like climatology, economics, engineering, genomics, hydrology, etc. (see Aue and Horváth, 2013; Jandhyala et al, 2013; Woodall and Montgomery, 2014; Sharma et al, 2016; Chakraborti and Graham, 2019; Truong et al, 2020, among many others). The errors (εi,n)i=1,...,n in model (1.1) are usually assumed to form at least a stationary process and many theoretical results for detecting multiple change points are only available for independent identically distributed error processes These assumptions simplify the statistical analysis of structural breaks. The currently available self-normalization procedures are based on partial sum processes (see Shao, 2015, for a recent review), which usually (under the assumption of stationarity) have a limiting process of the form {σW (λ)}λ∈[0,1], where {W (λ)}λ∈[0,1] is a known stochastic process and σ an unknown factor encapsulating the dependency structure of the underlying process In this case the factorisation of the limit into the long-run variance and a probabilistic term is used to construct a pivotal test statistic by forming a ratio such that the factor σ in the numerator and denominator cancels. We present a full self-normalization procedure for non-stationary time series, which might be useful for testing classical hypotheses
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
