7,141 research outputs found
Rare Probability Estimation under Regularly Varying Heavy Tails
This paper studies the problem of estimating the probability of symbols that have occurred very rarely, in samples drawn independently from an unknown, possibly infinite, discrete distribution. In particular, we study the multiplicative consistency of estimators, defined as the ratio of the estimate to the true quantity converging to one. We first show that the classical Good-Turing estimator is not universally consistent in this sense, despite enjoying favorable additive properties. We then use Karamata's theory of regular variation to prove that regularly varying heavy tails are sufficient for consistency. At the core of this result is a multiplicative concentration that we establish both by extending the McAllester-Ortiz additive concentration for the missing mass to all rare probabilities and by exploiting regular variation. We also derive a family of estimators which, in addition to being consistent, address some of the shortcomings of the Good-Turing estimator. For example, they perform smoothing implicitly and have the absolute discounting structure of many heuristic algorithms. This also establishes a discrete parallel to extreme value theory, and many of the techniques therein can be adapted to the framework that we set forth.National Science Foundation (U.S.) (Grant 6922470)United States. Office of Naval Research (Grant 6918937
Efficient rare-event simulation for the maximum of heavy-tailed random walks
Let be a sequence of i.i.d. r.v.'s with negative mean. Set
and define . We propose an importance sampling
algorithm to estimate the tail of that is strongly
efficient for both light and heavy-tailed increment distributions. Moreover, in
the case of heavy-tailed increments and under additional technical assumptions,
our estimator can be shown to have asymptotically vanishing relative variance
in the sense that its coefficient of variation vanishes as the tail parameter
increases. A key feature of our algorithm is that it is state-dependent. In the
presence of light tails, our procedure leads to Siegmund's (1979) algorithm.
The rigorous analysis of efficiency requires new Lyapunov-type inequalities
that can be useful in the study of more general importance sampling algorithms.Comment: Published in at http://dx.doi.org/10.1214/07-AAP485 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Efficient Simulation and Conditional Functional Limit Theorems for Ruinous Heavy-tailed Random Walks
The contribution of this paper is to introduce change of measure based
techniques for the rare-event analysis of heavy-tailed stochastic processes.
Our changes-of-measure are parameterized by a family of distributions admitting
a mixture form. We exploit our methodology to achieve two types of results.
First, we construct Monte Carlo estimators that are strongly efficient (i.e.
have bounded relative mean squared error as the event of interest becomes
rare). These estimators are used to estimate both rare-event probabilities of
interest and associated conditional expectations. We emphasize that our
techniques allow us to control the expected termination time of the Monte Carlo
algorithm even if the conditional expected stopping time (under the original
distribution) given the event of interest is infinity -- a situation that
sometimes occurs in heavy-tailed settings. Second, the mixture family serves as
a good approximation (in total variation) of the conditional distribution of
the whole process given the rare event of interest. The convenient form of the
mixture family allows us to obtain, as a corollary, functional conditional
central limit theorems that extend classical results in the literature. We
illustrate our methodology in the context of the ruin probability , where is a random walk with heavy-tailed increments that have
negative drift. Our techniques are based on the use of Lyapunov inequalities
for variance control and termination time. The conditional limit theorems
combine the application of Lyapunov bounds with coupling arguments
On the residual dependence index of elliptical distributions
The residual dependence index of bivariate Gaussian distributions is
determined by the correlation coefficient. This tail index is of certain
statistical importance when extremes and related rare events of bivariate
samples with asymptotic independent components are being modeled. In this paper
we calculate the partial residual dependence indices of a multivariate
elliptical random vector assuming that the associated random radius is in the
Gumbel max-domain of attraction. Furthermore, we discuss the estimation of
these indices when the associated random radius possesses a Weibull-tail
distribution.Comment: 11 pages, case \theta=1 now include
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