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