On the Laplace Transform of the Lognormal Distribution

Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyse a closed-form approximat...

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Veröffentlicht in:	Methodology and computing in applied probability Jg. 18; H. 2; S. 441 - 458
Hauptverfasser:	Asmussen, Søren, Jensen, Jens Ledet, Rojas-Nandayapa, Leonardo
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.06.2016 Springer Nature B.V
Schlagworte:	Approximation Asymptotic properties Business and Management Distribution Economics Electrical Engineering Exact solutions Laplace transforms Life Sciences Mathematical analysis Mathematical models Mathematics and Statistics Monte Carlo simulation Probability Statistics Studies Texts United States > US Monte Carlo method Lambert W function 90-04 60E10 Rare event simulation Laplace’s method Lognormal distribution Characteristic function Efficiency Laplace transform Importance sampling Moment generating function 60E05
ISSN:	1387-5841, 1573-7713
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Abstract	Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyse a closed-form approximation ℒ ~ ( 𝜃 ) of the Laplace transform ℒ ( 𝜃 ) which is obtained via a modified version of Laplace’s method. This approximation, given in terms of the Lambert W (⋅) function, is tractable enough for applications. We prove that ~( 𝜃 ) is asymptotically equivalent to ℒ( 𝜃 ) as 𝜃 → ∞ . We apply this result to construct a reliable Monte Carlo estimator of ℒ( 𝜃 ) and prove it to be logarithmically efficient in the rare event sense as 𝜃 → ∞ .
AbstractList	Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyse a closed-form approximation ℒ ~ ( 𝜃 ) of the Laplace transform ℒ ( 𝜃 ) which is obtained via a modified version of Laplace’s method. This approximation, given in terms of the Lambert W (⋅) function, is tractable enough for applications. We prove that ~( 𝜃 ) is asymptotically equivalent to ℒ( 𝜃 ) as 𝜃 → ∞ . We apply this result to construct a reliable Monte Carlo estimator of ℒ( 𝜃 ) and prove it to be logarithmically efficient in the rare event sense as 𝜃 → ∞ . (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyse a closed-form approximation ... of the Laplace transform ... which is obtained via a modified version of Laplace's method. This approximation, given in terms of the Lambert W() function, is tractable enough for applications. We prove that ~([eth]oef) is asymptotically equivalent to ([eth]oef) as [eth]oef arrow right infinity . We apply this result to construct a reliable Monte Carlo estimator of ([eth]oef) and prove it to be logarithmically efficient in the rare event sense as [eth]oef arrow right infinity . (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide variety of methods have been employed to provide approximations, both analytical and numerical. In this paper, we analyse a closed-form approximation ... of the Laplace transform ... which is obtained via a modified version of Laplace's method. This approximation, given in terms of the Lambert W() function, is tractable enough for applications. We prove that ~(ðoeoef) is asymptotically equivalent to (ðoeoef) as ðoeoef [arrow right] ∞. We apply this result to construct a reliable Monte Carlo estimator of (ðoeoef) and prove it to be logarithmically efficient in the rare event sense as ðoeoef [arrow right] ∞.
Author	Rojas-Nandayapa, Leonardo Asmussen, Søren Jensen, Jens Ledet
Author_xml	– sequence: 1 givenname: Søren surname: Asmussen fullname: Asmussen, Søren organization: Department of Mathematics, Aarhus University – sequence: 2 givenname: Jens Ledet surname: Jensen fullname: Jensen, Jens Ledet organization: Department of Mathematics, Aarhus University – sequence: 3 givenname: Leonardo surname: Rojas-Nandayapa fullname: Rojas-Nandayapa, Leonardo email: l.rojasnandayapa@uq.edu.au organization: School of Mathematics and Physics, University of Queensland
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Cites_doi	10.1007/978-1-4419-9473-8 10.1109/TVT.2006.878610 10.1080/03610928908830173 10.1115/1.3625776 10.1007/BF02124750 10.1017/CBO9780511619083 10.1109/TCOMM.2010.05.080640 10.1364/JOSA.66.000211 10.1017/S0334270000006901 10.1239/jap/1214950365 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2 10.1109/TVT.2004.823494 10.1111/j.2517-6161.1963.tb00521.x 10.1007/978-0-387-69033-9
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Keywords	Monte Carlo method Lambert W function 90-04 60E10 Rare event simulation Laplace’s method Lognormal distribution Characteristic function Efficiency Laplace transform Importance sampling Moment generating function 60E05
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References	BeaulieuNCXieQAn optimal lognormal approximation to lognormal sum distributionsIEEE Trans Veh Technol200453247948910.1109/TVT.2004.823494 GubnerJAA new formula for lognormal characteristic functionsIEEE Trans Veh Technol20065551668167110.1109/TVT.2006.878610 AitchisonIBrownJACThe lognormal distribution with special reference to its uses in economics1957CambridgeCambridge University Press0081.14303 Asmussen S, Jensen JL, Rojas-Nandayapa L (2014) Exponential family techniques for the lognormal left tail. Research Report LeipnikRBOn lognormal random variables: I. the characteristic functionJ Aust Math Soc Ser B199132327347107946210.1017/S03342700000069010733.60033 BarakatRSums of independent lognormally distributed random variablesJ Opt Soc Am19766621121642347410.1364/JOSA.66.000211 Barouch E, Kaufman GM (1976) On sums of lognormal random variables. Working paper. A. P. Sloan School of Management, MIT HeydeCOn a property of the lognormal distributionJ Roy Stat Soc Ser B1963293923931713360114.33802 de Bruijn NG (1970) Asymptotic methods in analysis. Courier Dover Publications HolgatePThe lognormal characteristic functionCommun Stat-Theor Methods19891845394548104672610.1080/036109289088301730721.60012 Rojas-Nandayapa L (2008) Risk probabilities: asymptotics and simulation. Ph.D. thesis, Aarhus University AsmussenSGlynnPWStochastic simulation: algorithms and analysis2007New YorkSpringer-Verlag1126.65001 Butler RW (2007) Saddlepoint approximations with applications. Cambridge University Press CorlessRMGonnetGHHareDEGJeffreyDJKnuthDEOn the Lambert W functionAdv Comput Math19965329359141428510.1007/BF021247500863.65008 CrowELShimizuKLognormal distributions: theory and applications1988New YorkMarcel Dekker Inc.0644.62014 Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1, 2nd edn. Wiley, New York Small CG (2013) Expansions and asymptotics for statistics. Chapman & Hall/CRC Dufresne D (2008) Sums of lognormals. In: Actuarial research conference proceedings LimpertEStahelWAAbbtMLog-normal distributions across the sciences: keys and cluesBioScience20015134135210.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2 RossbergAGLaplace transforms or probability distributions and their inversions are easy on logarithmic scalesJ Appl Probab200845531541242684910.1239/jap/12149503651154.65004 JensenJLSaddlepoint approximations1994OxfordOxford Science Publications1274.62008 TellamburaCSeranarteDAccurate computation of the MGF of the lognormal distribution and its application to sum of lognormalsIEEE Trans Commun2010581568157710.1109/TCOMM.2010.05.080640 Abramowitz M, Stegun I (1964) Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Applied mathematics series. Dover Publications Foss S, Korshunov D, Zachary S (2011) An introduction to heavy-tailed and subexponential distributions. Springer I Aitchison (9430_CR2) 1957 9430_CR4 S Asmussen (9430_CR3) 2007 9430_CR13 JL Jensen (9430_CR17) 1994 9430_CR1 NC Beaulieu (9430_CR7) 2004; 53 9430_CR18 R Barakat (9430_CR5) 1976; 66 EL Crow (9430_CR10) 1988 9430_CR11 JA Gubner (9430_CR14) 2006; 55 9430_CR12 9430_CR23 C Heyde (9430_CR15) 1963; 29 9430_CR21 E Limpert (9430_CR20) 2001; 51 AG Rossberg (9430_CR22) 2008; 45 P Holgate (9430_CR16) 1989; 18 9430_CR8 RM Corless (9430_CR9) 1996; 5 9430_CR6 RB Leipnik (9430_CR19) 1991; 32 C Tellambura (9430_CR24) 2010; 58
References_xml	– reference: Asmussen S, Jensen JL, Rojas-Nandayapa L (2014) Exponential family techniques for the lognormal left tail. Research Report – reference: Butler RW (2007) Saddlepoint approximations with applications. Cambridge University Press – reference: JensenJLSaddlepoint approximations1994OxfordOxford Science Publications1274.62008 – reference: AsmussenSGlynnPWStochastic simulation: algorithms and analysis2007New YorkSpringer-Verlag1126.65001 – reference: BeaulieuNCXieQAn optimal lognormal approximation to lognormal sum distributionsIEEE Trans Veh Technol200453247948910.1109/TVT.2004.823494 – reference: CorlessRMGonnetGHHareDEGJeffreyDJKnuthDEOn the Lambert W functionAdv Comput Math19965329359141428510.1007/BF021247500863.65008 – reference: Small CG (2013) Expansions and asymptotics for statistics. Chapman & Hall/CRC – reference: LeipnikRBOn lognormal random variables: I. the characteristic functionJ Aust Math Soc Ser B199132327347107946210.1017/S03342700000069010733.60033 – reference: HeydeCOn a property of the lognormal distributionJ Roy Stat Soc Ser B1963293923931713360114.33802 – reference: Foss S, Korshunov D, Zachary S (2011) An introduction to heavy-tailed and subexponential distributions. 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Snippet	Integral transforms of the lognormal distribution are of great importance in statistics and probability, yet closed-form expressions do not exist. A wide... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Integral transforms of the lognormal distribution are of great importance in... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).Integral transforms of the lognormal distribution are of great importance in...
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SubjectTerms	Approximation Asymptotic properties Business and Management Distribution Economics Electrical Engineering Exact solutions Laplace transforms Life Sciences Mathematical analysis Mathematical models Mathematics and Statistics Monte Carlo simulation Probability Statistics Studies Texts
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Title	On the Laplace Transform of the Lognormal Distribution
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