Electronic Thesis and Dissertation Repository


Doctor of Philosophy


Statistics and Actuarial Sciences


A. I. McLeod


Persistent and anti-persistent time series processes show what is called hyperbolic decay. Such series play an important role in the study of many diverse areas such as geophysics and financial economics. They are also of theoretical interest. Fractional Gaussian noise (FGN) and fractionally-differeneced white noise are two widely known examples of time series models with hyperbolic decay. New closed form expressions are obtained for the spectral density functions of these models. Two lesser known time series models exhibiting hyperbolic decay are introduced and their basic properties are derived. A new algorithm for approximate likelihood estimation of the models using frequency domain methods is derived and implemented in R. The issue of mean estimation and multimodality in time series, particularly in the simple case of one short memory component and one hyperbolic component is discussed. Methods for visualizing bimodal surfaces are discussed. The exact prediction variance is derived for any model that admits an autocovariance function and integrated (inverse-differenced) by integer d. A new software package in R, arfima, for exact simulation, estimation, and forecasting of mixed short-memory and hyperbolic decay time series. This package has a wider functionality and increased reliability over other software that is available in R and elsewhere.