Efficient simulation of tail probabilities of sums of dependent random variables

Blanchet, José; Rojas-Nandayapa, Leonardo

doi:10.1017/s0021900200099198

Cited by 5 publications

(2 citation statements)

References 19 publications

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“…However, the Fenton–Wilkinson method, being a central limit type result, can deliver rather inaccurate approximations of the distribution of the lognormal sum when the number of summands is rather small or when the dispersion parameter is too high—in particular in the tail regions. Another topic, which has been much studied recently, is approximations and simulation algorithms for right tail probabilities

double-struckP (S_{n} \geq y)

under heavy‐tailed assumptions and allowing for dependence; see in particular Asmussen & Rojas‐Nandayapa (); Foss & Richards (); Mitra & Resnick (); Asmussen et al (); Blanchet & Rojas‐Nandayapa (). For further literature surveys, see Gulisashvili & Tankov ().…”

Section: Introductionmentioning

confidence: 99%

Exponential Family Techniques for the Lognormal Left Tail

Asmussen

Jensen

Rojas-Nandayapa

2016

Scandinavian J Statistics

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Let X be lognormal(μ,σ2) with density f(x); let θ > 0 and define L(θ)=double-struckEnormale−θX. We study properties of the exponentially tilted density (Esscher transform) fθ(x) = e−θxf(x)/L(θ), in particular its moments, its asymptotic form as θ→∞ and asymptotics for the saddlepoint θ(x) determined by double-struckE[Xnormale−θX]/L(θ)=x. The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of the sum of lognormals Sn=X1+⋯+Xn: a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf Fn(x) and the pdf fn(x) of Sn are given in a range of values of σ2,n and x motivated by portfolio value‐at‐risk calculations.

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double-struckP (S_{n} \geq y)

Section: Introductionmentioning

confidence: 99%

Exponential Family Techniques for the Lognormal Left Tail

Asmussen

Jensen

Rojas-Nandayapa

2016

Scandinavian J Statistics

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“…The optimal mixture parameters are chosen by solving a limiting control problem. With the exception of [24,7] which consider correlated sums, it is instructive to note that most IS methods involving a number of heavy-tailed random variables are applicable only when the random variables involved are mutually independent. A strategy for handling heavy-tailed random vectors, in the context of scenario generation for chance constraints, is developed in [22].…”

Section: 1mentioning

confidence: 99%

Achieving Efficiency in Black Box Simulation of Distribution Tails with Self-structuring Importance Samplers

Deo¹,

Murthy²

2021

Preprint

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Motivated by the increasing adoption of models which facilitate greater automation in risk management and decision-making, this paper presents a novel Importance Sampling (IS) scheme for measuring distribution tails of objectives modeled with enabling tools such as feature-based decision rules, mixed integer linear programs, deep neural networks, etc. Conventional efficient IS approaches suffer from feasibility and scalability concerns due to the need to intricately tailor the sampler to the underlying probability distribution and the objective. This challenge is overcome in the proposed black-box scheme by automating the selection of an effective IS distribution with a transformation that implicitly learns and replicates the concentration properties observed in less rare samples. This novel approach is guided by a large deviations principle that brings out the phenomenon of self-similarity of optimal IS distributions. The proposed sampler is the first to attain asymptotically optimal variance reduction across a spectrum of multivariate distributions despite being oblivious to the underlying structure. The large deviations principle additionally results in new distribution tail asymptotics capable of yielding operational insights. The applicability is illustrated by considering product distribution networks and portfolio credit risk models informed by neural networks as examples.

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Efficient Algorithms for Tail Probabilities of Exchangeable Lognormal Sums

Dingec

Hörmann

2021

Methodol Comput Appl Probab

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Efficient simulation of tail probabilities of sums of dependent random variables

Cited by 5 publications

References 19 publications

Exponential Family Techniques for the Lognormal Left Tail

Exponential Family Techniques for the Lognormal Left Tail

Achieving Efficiency in Black Box Simulation of Distribution Tails with Self-structuring Importance Samplers

Efficient Algorithms for Tail Probabilities of Exchangeable Lognormal Sums

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