Date of Award


Degree Type


Degree Name

Doctor of Philosophy


Statistics and Actuarial Sciences


Dr. Ian McLeod


An overview of the thesis is given in Chapter 1. Chapter 2 discusses a new efficient methodology for computing the autocovariance function of a class of fractional ARMA models and the application of this technique in an algorithm for exact maximum likelihood estimation. A new statistical method for comparing the goodness-of-fit using the relative plausibity based on the AIC or BIC is introduced. Comparison of goodness-of-fit among two types of hyperbolic decay models and standard ARMA models are examined with twenty one annual geophysical datasets. Although it is shown that most annual series do not conclusively exhibit long-memory, several cases where the long-memory model gives a statistically significant better fit than the standard ARMA model are noted. Chapter 3 introduces a new hyperbolic decay time series model for long memory time series. Asymptotic properties of this model are derived. The goodness-of-fit of the new model is compared with existing hyperbolic decays models. Chapter 4 provides a new technique to derive the exact Fisher information matrix for irregular ARMA models. Chapter 5 discusses the visualization of space-time data observed over an irregular area. Kriging is used to interpolate the data on a regular grid. Then data is then visualized using animation and multipanel displays. It is demonstrated with actual storm data how these visualizations facilitate comparison of values, trends and relationships in the spatial- time series. Chapter 6 uses the concept of stability to study the behaviour of a generalized nonlinear ARCH time series model. A necessary and sufficient condition of stability is derived. Supplementary material including data, programs and visualizations are available from the thesis website at http://www.stats.uwo.ca/mcleod/projects/lam/.

Keywords: Autocovariance function, fractional ARMA, hyperbolic decay models, iii irregular ARMA, spatial-time series, long memory, exact Fisher information matrix, stability, and visualizations.

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