Detecting gradual changes in locally stationary processes

Abstract

In a wide range of applications, the stochastic properties of the observed time series change over time. The changes often occur gradually rather than abruptly: the properties are (approximately) constant for some time and then slowly start to change. In many cases, it is of interest to locate the time point where the properties start to vary. In contrast to the analysis of abrupt changes, methods for detecting smooth or gradual change points are less developed and often require strong parametric assumptions. In this paper, we develop a fully nonparametric method to estimate a smooth change point in a locally stationary framework. We set up a general procedure which allows us to deal with a wide variety of stochastic properties including the mean, (auto)covariances and higher moments. The theoretical part of the paper establishes the convergence rate of the new estimator. In addition, we examine its finite sample performance by means of a simulation study and illustrate the methodology by two applications to financial return data.Comment: Published at http://dx.doi.org/10.1214/14-AOS1297 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotics for change-point models under varying degrees of mis-specification

“…In all these papers, the limit of the sequence of change–point models considered–the so-called ‘null’ model – is a smooth function without a change–point, whereas we have the reverse scenario: our sequence of models are smooth functions that, in the limit, produce a discontinuous change–point model. Our work is also quite different from inference in settings where the change-point is not a discontinuity but represents a point of smooth and/or gradual change; see, for example, Vogt and Dette (2015). To the best of our knowledge, our work is the first attempt at providing a comprehensive as well as systematic understanding of the behavior of change-point models under local smooth alternatives.…”

On the Estimation of Locally Stationary Long-Memory Processes

“…A large number of estimation and hypothesis testing methods have been developed from these seminal ideas, see for example, Chandler and Polonik (2017), Paparoditis and Preuss (2015), Guinness and Fuentes (2015), Chen et al (2018), Fiecas and Ombao (2016), Song et al (2016), Wu and Zhou (2011), Puchstein and Preuss (2016), Rosen et al (2012), Vogt and Dette (2015), Kreiss and Paparoditis (2015), , Zhou (2014), Nason (2013), Preuss et al (2013b) Guinness and Stein (2013), Giraitis et al (2014), Preuss et al (2013a), Zhou (2013), Roueff and Von Sachs (2011), Dette et al (2011), Van Bellegem and Dahlhaus (2006) and Beran (2009), among others.…”

Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes

“…Most authors consider nonparametric location or parametric models with independently distributed observations and we refer to Bissell (1984), Gan (1991), Siegmund and Zhang (1994), Husková (1999), Husková and Steinebach (2002) and Mallik et al (2013) among others [see also Aue and Steinebach (2002) for some results in a time series model]. Recently Vogt and Dette (2015) developed a nonparametric method to estimate a change point corresponding to a smooth change of a locally stationary time series, and the present paper is devoted to the development of nonparametric inference for gradual changes in the jump properties of a discretely observed Itō semimartingale.…”

“…In Section 2 we introduce the formal setup as well as a measure of time variation which is similar to Vogt and Dette (2015) and used to identify changes in the jump characteristic later on. Section 3 is concerned with weak convergence of an estimator for this measure, and as a consequence we also obtain weak convergence of related statistics which can be used for testing for a gradual change and for localizing the first change point.…”

“…In order to investigate gradual changes in the jump behaviour of the underlying process we follow Vogt and Dette (2015) and consider a measure of time variation for the jump behaviour, which is defined by…”