Exponential Family Techniques for the Lognormal Left Tail



Asmussen, S, Jensen, JL and Rojas-Nandayapa, L ORCID: 0000-0001-5652-3183
(2016) Exponential Family Techniques for the Lognormal Left Tail. Scandinavian Journal of Statistics: Theory and Applications, 43 (3). pp. 774-787.

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Abstract

Let X be lognormal(μ,σ2) with density f(x); let θ > 0 and define urn:x-wiley:sjos:media:sjos12203:sjos12203-math-0001. 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 urn:x-wiley:sjos:media:sjos12203:sjos12203-math-0002. 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.

Item Type: Article
Uncontrolled Keywords: Cramér function, Esscher transform, exponential change of measure, importance sampling, Lambert W function, Laplace method, Laplace transform, lognormal distribution, outage probability, rare event simulation, saddlepoint approximation, VaR
Depositing User: Symplectic Admin
Date Deposited: 31 May 2017 08:25
Last Modified: 19 Jan 2023 07:03
DOI: 10.1111/sjos.12203
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3007730