Characterization and estimation of rare events via extreme values.

摘要

We consider the problem of characterizing and monitoring rare events, where the main vehicle in the analysis is extreme value theory. Although the methodology we develop and investigate applies in several areas, our exposition is ultimately concerned with queueing related processes, and systems in which congestion and backlog effects are prevalent.; This dissertation consists of three closely related yet self-contained essays. The first part deals with the problem of estimating tail probabilities in queues, using a semi-parametric approach. We introduce an “extremal” estimator, based on the maximum value of the workload in the system. The main results indicate that in order to successfully estimate and extrapolate buffer overflow probabilities, it is in some sense necessary to first introduce a rough model for the behavior of the tails of the marginal distribution. In the course of this investigation, we establish new limit theory in the general context of extreme values and first passage times in regenerative processes.; The second part of the thesis considers the problem of estimating the tail decay parameter of the marginal distribution corresponding to a generic discrete-time, real-valued stationary stochastic process. These estimators are based on the maximal extreme value of the process observed over a sampled time interval. Consistency and robustness properties of these estimators are established, and a moving average variant of the aforementioned estimator is also investigated.; The third, and final part of the dissertation focuses on a particular example of a queue fed by fractional Brownian motion (fBM). When the queue is stable, we prove that rate at which the maximum of the workload process grows over time is more rapid compared with the case where input processes are light-tailed and short-range dependent. Consequently, one also has that the typical time required to reach a large buffer level is shorter. While this has obvious ramifications and implications on system design, we discuss one particular consequence: statistical estimation of the tail probabilities associated with the steady-state workload distribution.