| Many sources of data have abrupt changes in distribution, called change-points by statisticians. Shocks to systems (such as currency crisis on financial markets), or abrupt changes of state (such as the arrival of a new shockwave during an earthquake) can be the underlying cause of these change-points. In other situations, such as when detecting edges in images, models that allow for discontinuities better fit the observed data. Most methods that have been developed to look for change-points are characterized by the mean of a function. Those that look for the abrupt changes in the variance usually assume that the variance is constant between change-points.; This dissertation makes several novel contributions to the field. We model our series as being locally stationary between change-points. Nearby observations have similar distributions, but can differ, even away from change-points. We start by illustrating how a locally linear estimate, developed to detect abrupt changes in the mean, can be adapted to detect abrupt changes in the variance, assuming independent observations and a continuous variance function between change-points. We then consider the problem of detecting abrupt changes in the variance structure when the data can be dependent. We find that the symmetrized Kullback-Leibler discrimination information provides a good distance measure in this setting, and we explore its asymptotic properties. We also illustrate that it out-performs a variety of other distance measures on various simulated series, and explore where it fails.; Finite-sample distributions are difficult for our test statistics, so we explore the performance of its bootstrap. For the locally-linear estimate, a bootstrap developed by Gijbels and Goderniaux is available. For the symmetrized Kullback-Leibler discrimination information, several bootstraps already described in the literature for stationary series were adapted for our locally-stationary needs, and were compared with each other. We find both a parametric and a non-parametric bootstrap with similar performance.; We also present some initial work in a new research project we propose that applies the ideas developed in this dissertation for an automated speech recognition system. |
