Modelling and Estimation of Time Series with Long Memory

Many real-world time series have been observed to have strong positive correlation between their long-term observed values, and this behaviour is known as long memory or long-range dependence. However, many statistical models and estimation techniques are built under the assumption of short memory (or sometimes complete independence) and identically distributed data. This challenges the modelling and estimation of time series with long memory. In the first half of this thesis, we investigate several parametric models for time series with long memory, which are commonly known as fractional models. We focus on the challenge of parameter estimation from sampled time series, and compare numerous existing and novel methods in wide-ranging simulation studies. The estimation results from all these methods are provided and compared among each other to argue that a novel method, known as the debiased Whittle likelihood estimator, originally proposed for time series with short memory, is also the most appropriate method for long memory in terms of mean squared error, consistency, asymptotic efficiency, and computational cost. We then implement this estimator to study long-memory behaviour found in real-world financial time series, as an example. In the second half of this thesis, we investigate two other applications of time series with long-memory behaviour in health sciences and sport sciences. Both applications are concerned with high-frequency tracking of the movement of individuals, the former concerned with monitoring activity levels of patients with advanced dementia, and the latter concerned with footballers' movements during professional matches. In both applications, the observed time series exhibit cycles, trends, and changes in variability, as well as long-memory behaviour. Therefore parametric models are difficult to construct for the time series of these applications, and we instead propose nonparametric measures providing summary statistics jointly capturing long-memory behaviour alongside other summary statistics of interest