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Rare events simulation for heavy-tailed distributions. (English) Zbl 0958.65010
The authors consider the problem of simulation of rare events. The family \(\{ A(x)\}\) of events defined on a probability space \(\{ \Omega, F,P\}\) are rare in the sense that \(z(x)=P(A(x))\to 0\), as \(x\to\infty.\) An estimator for \(z(x)\) is a random variable \(Z(x)\) such that \(z(x)=E Z(x).\) The simulation is performed by producing \(N\) independent and identically distributed random variables \(Z_1, \ldots, Z_n\) such that \(EZ_i =z(x)\) and form the estimate of \(z(x)\) \((Z_1 +\cdots + Z_N)/N\), and a confidence interval based upon the empirical variance of the \(Z_i.\) There exist standard algorithms for the simulation for the light-tailed case that is when the distributions have the exponential moments.
The authors consider heavy-tailed distributions, for example, log-normal, Weibull with decreasing failure rate \(1-G(x)=\exp\{-x^\beta\}\) with \(0<\beta<1\), regulary varying, \(1-G(x)= L(x)/x^\alpha\), where \(L\) is a slowly varying function. The authors present two asymptotically efficient algorithms which use a conditional Monte Carlo method and order statistics, and method using importance sampling idea.

MSC:
65C50 Other computational problems in probability (MSC2010)
60F10 Large deviations
65C05 Monte Carlo methods
65C60 Computational problems in statistics (MSC2010)
62G30 Order statistics; empirical distribution functions
60G50 Sums of independent random variables; random walks
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