264 research outputs found
Asymptotic optimality of the cross-entropy method for Markov chain problems
The correspondence between the cross-entropy method and the zero-variance
approximation to simulate a rare event problem in Markov chains is shown. This
leads to a sufficient condition that the cross-entropy estimator is
asymptotically optimal.Comment: 13 pager; 3 figure
Monte Carlo Methods for Insurance Risk Computation
In this paper we consider the problem of computing tail probabilities of the
distribution of a random sum of positive random variables. We assume that the
individual variables follow a reproducible natural exponential family (NEF)
distribution, and that the random number has a NEF counting distribution with a
cubic variance function. This specific modelling is supported by data of the
aggregated claim distribution of an insurance company. Large tail probabilities
are important as they reflect the risk of large losses, however, analytic or
numerical expressions are not available. We propose several simulation
algorithms which are based on an asymptotic analysis of the distribution of the
counting variable and on the reproducibility property of the claim
distribution. The aggregated sum is simulated efficiently by importancesampling
using an exponential cahnge of measure. We conclude by numerical experiments of
these algorithms.Comment: 26 pages, 4 figure
Importance Sampling Simulations of Markovian Reliability Systems using Cross Entropy
This paper reports simulation experiments, applying the cross entropy method suchas the importance sampling algorithm for efficient estimation of rare event probabilities in Markovian reliability systems. The method is compared to various failurebiasing schemes that have been proved to give estimators with bounded relativeerrors. The results from the experiments indicate a considerable improvement ofthe performance of the importance sampling estimators, where performance is mea-sured by the relative error of the estimate, by the relative error of the estimator,and by the gain of the importance sampling simulation to the normal simulation
Large Deviations Methods and the Join-the-Shortest-Queue Model
We develop a methodology for studying ''large deviations type'' questions. Our approach does not require that the large deviations principle holds, and is thus applicable to a larg class of systems. We study a system of queues with exponential servers, which share an arrival stream. Arrivals are routed to the (weighted) shortest queue. It is not known whether the large deviations principle holds for this system. Using the tools developed here we derive large deviations type estimates for the most likely behavior, the most likely path to overflow and the probability of overflow. The analysis applies to any finite number of queues. We show via a counterexample that this sytem may exhibit unexpected behavior
Finite-state Markov Chains obey Benford's Law
A sequence of real numbers (x_n) is Benford if the significands, i.e. the
fraction parts in the floating-point representation of (x_n) are distributed
logarithmically. Similarly, a discrete-time irreducible and aperiodic
finite-state Markov chain with probability transition matrix P and limiting
matrix P* is Benford if every component of both sequences of matrices (P^n -
P*) and (P^{n+1}-P^n) is Benford or eventually zero. Using recent tools that
established Benford behavior both for Newton's method and for
finite-dimensional linear maps, via the classical theories of uniform
distribution modulo 1 and Perron-Frobenius, this paper derives a simple
sufficient condition (nonresonant) guaranteeing that P, or the Markov chain
associated with it, is Benford. This result in turn is used to show that almost
all Markov chains are Benford, in the sense that if the transition
probabilities are chosen independently and continuously, then the resulting
Markov chain is Benford with probability one. Concrete examples illustrate the
various cases that arise, and the theory is complemented with several
simulations and potential applications.Comment: 31 pages, no figure
Large Deviations without Principle: Join the Shortest Queue
We develop a methodology for studying "large deviations type" questions. Our approach does not require that the large deviations principle holds, and is thus applicable to a larg class of systems. We study a system of queues with exponential servers, which share an arrival stream. Arrivals are routed to the (weighted) shortest queue. It is not known whether the large deviations principle holds for this system. Using the tools developed here we derive large deviations type estimates for the most likely behavior, the most likely path to overflow and the probability of overflow. The analysis applies to any finite number of queues. We show via a counterexample that this sytem may exhibit unexpected behavior
New exponential dispersion models for count data -- the ABM and LM classes
In their fundamental paper on cubic variance functions, Letac and Mora (The
Annals of Statistics,1990) presented a systematic, rigorous and comprehensive
study of natural exponential families on the real line, their characterization
through their variance functions and mean value parameterization. They
presented a section that for some reason has been left unnoticed. This section
deals with the construction of variance functions associated with natural
exponential families of counting distributions on the set of nonnegative
integers and allows to find the corresponding generating measures. As
exponential dispersion models are based on natural exponential families, we
introduce in this paper two new classes of exponential dispersion models based
on their results. For these classes, which are associated with simple variance
functions, we derive their mean value parameterization and their associated
generating measures. We also prove that they have some desirable properties.
Both classes are shown to be overdispersed and zero-inflated in ascending
order, making them as competitive statistical models for those in use in both,
statistical and actuarial modeling. To our best knowledge, the classes of
counting distributions we present in this paper, have not been introduced or
discussed before in the literature. To show that our classes can serve as
competitive statistical models for those in use (e.g., Poisson, Negative
binomial), we include a numerical example of real data. In this example, we
compare the performance of our classes with relevant competitive models.Comment: 27 pages, 4 tables, 3 figure
Capacity planning of prisons in the Netherlands
In this paper we describe a decision support system developed to help in assessing the need for various type of prison cells. In particular we predict the probability that a criminal has to be sent home because of a shortage of cells. The problem is modelled through a queueing network with blocking after service. We focus in particular on the new analytical method to solve
this network
W boson production at hadron colliders: the lepton charge asymmetry in NNLO QCD
We consider the production of W bosons in hadron collisions, and the
subsequent leptonic decay W->lnu_l. We study the asymmetry between the rapidity
distributions of the charged leptons, and we present its computation up to the
next-to-next-to-leading order (NNLO) in QCD perturbation theory. Our
calculation includes the dependence on the lepton kinematical cuts that are
necessarily applied to select W-> lnu_l events in actual experimental analyses
at hadron colliders. We illustrate the main differences between the W and
lepton charge asymmetry, and we discuss their physical origin and the effect of
the QCD radiative corrections. We show detailed numerical results on the charge
asymmetry in ppbar collisions at the Tevatron, and we discuss the comparison
with some of the available data. Some illustrative results on the lepton charge
asymmetry in pp collisions at LHC energies are presented.Comment: 37 pages, 21 figure
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