Exponential family techniques for the lognormal left tail

Asmussen, Soren, Jensen, Jens L. and Rojas-Nandayapa, Leonardo (2016) Exponential family techniques for the lognormal left tail. Scandinavian Journal of Statistics, 43 3: 774-787. doi:10.1111/sjos.12203

Author Asmussen, Soren
Jensen, Jens L.
Rojas-Nandayapa, Leonardo
Title Exponential family techniques for the lognormal left tail
Journal name Scandinavian Journal of Statistics   Check publisher's open access policy
ISSN 1467-9469
Publication date 2016
Year available 2016
Sub-type Article (original research)
DOI 10.1111/sjos.12203
Open Access Status Not Open Access
Volume 43
Issue 3
Start page 774
End page 787
Total pages 14
Place of publication Chichester, West Sussex, United Kingdom
Publisher Wiley-Blackwell Publishing
Collection year 2017
Language eng
Abstract Let X be lognormal(μ,σ2) with density f(x); let θ > 0 and define inline image. 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[RIGHTWARDS ARROW]∞ and asymptotics for the saddlepoint θ(x) determined by inline image. 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.
Keyword Cramér function
Esscher transform
Exponential change of measure
Importance sampling
Lambert W function
Laplace method
Laplace transform
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
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