Hong Liu Abstract
PhD, August 2006
Rare Events, Heavy Tails, and Simulation
Advisor: Jem Corcoran
We explore current developments for rare events simulation for stochastic processes where the rare event is caused by an underlying heavy-tailed distribution. These distributions fail to have the exponential moments required for standard existing algorithms that apply to light-tailed distributions. A specific problem concerning sojourn times in the generalized Jackson network is discussed. We present four existing algorithms that work well in very limited cases. These include the conditional Monte Carlo approach, the order statistics approach, and a transform likelihood-ratio method.
In this thesis we examine a long-standing conjecture about the tail distribution of sojourn times in the Jackson network. This conjecture turns out to be accurate only in certain cases and we propose an alternative for other situations.
Much of the research focus for heavy-tailed data is on its analysis and on the estimation of extreme value indices. A much smaller portion of work has been devoted to the detection of heavy tail in data. We develop a hypothesis involving max to sum ratios to test whether or not a data set was generated by an underlying heavy-tailed distribution. Our approach works well even for small sample sizes.
