Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science


Dr. Robert Lund

Committee Member

Dr. Peter Kiessler

Committee Member

Dr. Colin Gallagher

Committee Member

Dr. Brian Fralix


Extreme data points are important in environmental, financial, and insurance settings. In this work, we consider two topics on extremes from environmental data. Many environmental time series have a seasonal structure. The first part presents an approach to identify the rare events of such series based on time series residuals. Here, periodic autoregressive moving-average models are applied to describe the series. The methods justify the application of classical peaks over threshold methods to estimated versions of the one-step-ahead prediction errors of the series. Such methods enable the seasonal means, variances, and autocorrelations of the series to be taken into account. Even in stationary settings, the proposed strategy is useful as it bypasses the need for blocking runs of extremes. The mathematics are justified via a limit theorem for a periodic autoregressive moving-average time series. A detailed application to a daily temperature series from Griffin, Georgia, is pursued. In the second part, the asymptotic independence between sample means and maxima from a periodic time series is derived. The setup entails a causal periodic autoregressive moving-average model driven by IID periodic noise having a finite (2+δ) th moment. Our approach takes a regenerative process framework, truncating to reduce the analysis to a periodic moving-average model. The regenerative process is allowed to be partitioned into IID cycles. Here, point process techniques are used to quantify the distribution of the maximums and the Dé coupage de Lévy Theorem gives the asymptotic independence between maximums and partial sums for the periodic time series.