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Exponential Family Techniques for the Lognormal Left Tail
Author(s) -
Asmussen Søren,
Jensen Jens Ledet,
RojasNandayapa Leonardo
Publication year - 2016
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12203
Subject(s) - mathematics , logarithm , estimator , log normal distribution , exponential family , exponential function , range (aeronautics) , probability density function , combinatorics , statistics , mathematical analysis , materials science , composite material
Let X be lognormal( μ , σ 2 ) with density f ( x ); let θ > 0 and define L ( θ ) = E e − θX . We study properties of the exponentially tilted density (Esscher transform) f θ ( x ) = e − θ x f ( x )/ L ( θ ), in particular its moments, its asymptotic form as θ → ∞ and asymptotics for the saddlepoint θ ( x ) determined by E [ X e − 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 S n = X 1 +⋯+ X n : a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf F n ( x ) and the pdf f n ( x ) of S n are given in a range of values of σ 2 , n and x motivated by portfolio value‐at‐risk calculations.