Date of Award:

5-2006

Document Type:

Dissertation

Degree Name:

Doctor of Philosophy (PhD)

Department:

Mathematics and Statistics

Committee Chair(s)

Piotr Kokoszka

Committee

Piotr Kokoszka

Committee

Joseph Koebbe

Committee

Jürgen Symanzik

Committee

Stephen Yeung

Committee

Lie Zhu

Abstract

Several wavelet techniques in the analysis of time series are developed and applied to real data sets.

Methods for long-memory models include wavelet-based confidence intervals for the self-similarity parameter in potentially heavy-tailed observations. Empirical coverage probabilities are used to assess the procedures by applying them to Linear Fractional Stable Motion with many choices of parameters. Asymptotic confidence intervals provide empirical coverage often much lower than nominal and it is recommended to use subsampling confidence intervals. A procedure for monitoring the constancy of the self-similarity parameter is proposed and applied to Ethernet data sets.

A test to distinguish a weakly dependent time series with a trend component, from a long-memory process, possibly with a trend, is proposed. The test uses a generalized likelihood ratio statistic based on wavelet domain likelihoods. The test is robust to trends that are piecewise polynomials. The empirical size and power are good and do not depend on specific choices of wavelet functions and models for the wavelet coefficients. The test is applied to annual minima of the water levels of the Nile River and confirms the presence of long-range dependence in this time series.

A wavelet-based method of computing an index of geomagnetic storm activity is put forward. The new index can be computed automatically using statistical procedures and does not require operator's intervention in selecting quiet days and removal of the secular component by polynomial fitting. This one-minute index is designed to facilitate the study of the fine structure of geomagnetic storm events and requires only the most recent magnetogram records, e.g., the two months including the storm event of interest. It can thus be computed over a moving window as soon as new magnetogram records become available. Averaged over one-hour periods, it is practically indistinguishable from the traditional Dst index.

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