Meng, Xiangdong, authorKokoszka, Piotr S., advisorCooley, Dan, committee memberWang, Haonan, committee memberMiao, Hong, committee member2024-05-272024-05-272024https://hdl.handle.net/10217/238456In scalar time series analysis, a long-range dependent (LRD) series cannot be easily distinguished from certain non-stationary models, such as the change in mean model with short-range dependent (SRD) errors. To be specific, realizations of LRD series usually have a characteristic of changing local mean if the time span taken into account is long enough, which resembles the behavior of change in mean models. Test procedure for distinguishing between these two types of model has been investigated a lot in scalar case, see e.g. Berkes et al. (2006) and Baek and Pipiras (2012) and references therein. However, no analogous test for functional observations has been developed yet, partly because of omitted methods and theory for analyzing functional time series with long-range dependence. My dissertation establishes a procedure for testing change in mean models with SRD errors against LRD processes in functional case, which is an extension of the method of Baek and Pipiras (2012). The test builds on the local Whittle (LW) (or Gaussian semiparametric) estimation of the self-similarity parameter, which is based on the estimated level 1 scores of a suitable functional residual process. Remarkably, unlike other parametric methods such as Whittle estimation, whose asymptotic properties heavily depend on validity of the underlying spectral density on the full frequency range (−π, π], LW estimation imposes mild restrictions on the spectral density only near the origin and is thus more robust to model misspecification. We shall prove that the test statistic based on LW estimation is asymptotically normally distributed under the null hypothesis and it diverges to infinity under the LRD alternative.born digitaldoctoral dissertationsengCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.functional time serieschange pointlong-range dependenceText