A fast method to seek the mean and variance of the summation of lognormal variables

Authors

  • Juan Guillermo Torres Hurtado Universidad de los Andes https://orcid.org/0000-0001-8912-9289
  • Roberto Bustamante Universidad de los Andes
  • Carlos E. Caicedo Syracuse University

Keywords:

Computer applications, random processes, simulation techniques, statistical analysis

Abstract


The sum of lognormal variables has been a topic of interest in several fields of research such as engineering, biology and finance, among others. For example, in the field of telecommunications, the aggregate interference of radio frequency signals is modeled as a sum of lognormal variables. To date, there is no closed expression for the probability distribution function (PDF) of this sum. Several authors have proposed approximations for this PDF, with which they calculate the mean and variance. However, each method has limitations in its range of parameters for mean, variance and number of random variables to be added. In other cases, long approximations as power series are used, which makes the analytical treatment impractical and reduces the computational performance of numerical operations.  This paper shows an alternative method for calculating the mean and variance of the sum of lognormal random variables from a computational performance approach. Our method has been evaluated extensively by Monte Carlo simulations. As a result, this method is computationally efficient and yields a low approximation error computation for a wide range of mean values, variances and number of random variables.

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Author Biographies

Juan Guillermo Torres Hurtado, Universidad de los Andes

Electronic and electrical engineering. Student

Roberto Bustamante, Universidad de los Andes

Electronic and electrical engineering. Associate professor

Carlos E. Caicedo, Syracuse University

School of Information Studies, Associate professor

References

S. Nadarajah, “A Review of Results on Sums of Random Variables,” Acta Applicandae Mathematicae, vol. 103, no. 2, pp. 131–140, sep 2008.

G. L. Stuber, Principles of mobile communication. Springer, 2011.

E. Limpert, W. A. Stahel, and M. Abbt, “Log-normal Distributions across the Sciences: Keys and Clues,” BioScience, vol. 51, no. 5, p. 341, 2001.

D. Dufresne, “Sums of lognormals,” Society of actuaries, University of Melbourne, 2009.

A. Messica, “A simple low-computation-intensity model for approximating the distribution function of a sum of non-identical lognormals for financial applications,” in AIP Conference Proceedings, vol. 1773, 2016.

N. A. Marlow, “A Normal Limit Theorem for Power Sums of Independent Random Variables,” Bell System Technical Journal, vol. 46, no. 9, pp. 2081–2089, nov 1967.

L. Fenton, “The Sum of Log-Normal Probability Distributions in Scatter Transmission Systems,” IEEE Transactions on Communications, vol. 8, no. 1, pp. 57–67, 1960.

S. C. Schwartz and Y. S. Yeh, “On the Distribution Function and Moments of Power Sums With Log-Normal Components,” Bell System Technical Journal, vol. 61, no. 7, pp. 1441–1462, sep 1982.

E. L. C. K. Shimizu, Lognormal Distributions: Theory and Application. New York: Dekker, 1988.

A. Safak, “Statistical analysis of the power sum of multiple correlated log-normal components,” IEEE Transactions on Vehicular Technology, vol. 42, no. 1, pp. 58–61, 1993.

N. Beaulieu and Q. Xie, “An Optimal Lognormal Approximation to Lognormal Sum Distributions,” IEEE Transactions on Vehicular Technology, vol. 53, no. 2, pp. 479–489, mar 2004.

N. Beaulieu and F. Rajwani, “Highly Accurate Simple Closed-Form Approximations to Lognormal Sum Distributions and Densities,” IEEE Communications Letters, vol. 8, no. 12, pp. 709–711, dec 2004.

S. S. Szyszkowicz and H. Yanikomeroglu, “On the Tails of the Distribution of the Sum of Lognormals,” in 2007 IEEE International Conference on Communications. IEEE, jun 2007, pp. 5324–5329.

N. Mehta, J. Wu, A. Molisch, and J. Zhang, “Approximating a Sum of Random Variables with a Lognormal,” IEEE Transactions on Wireless Communications, vol. 6, no. 7, pp. 2690–2699, jul 2007.

C. Lam and T. Le-Ngoc, “Log-shifted gamma approximation to lognormal sum distributions,” IEEE Transactions on Vehicular Technology, vol. 56, no. 4 II, 2007.

A. S. H. Mahmoud and A. H. Rashed, “Efficient Computation of Distribution Function for Sum of Large Number of Lognormal Random Variables,” Arabian Journal for Science and Engineering, vol. 39, no. 5, pp. 3953–3961, may 2014.

J. Gubner, “A New Formula for Lognormal Characteristic Functions,” IEEE Transactions on Vehicular Technology, vol. 55, no. 5, pp. 1668–1671, sep 2006.

B. Wang, G. Cui, W. Yi, L. Kong, and X. Yang, “Approximation to independent lognormal sum with α-μ Distribution and the application,” Signal Processing, vol. 111, 2015.

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Published

2021-08-19

How to Cite

Torres Hurtado, J. G., Bustamante, R., & Caicedo, C. E. . (2021). A fast method to seek the mean and variance of the summation of lognormal variables. Revista Facultad De Ingeniería Universidad De Antioquia. Retrieved from https://revistas.udea.edu.co/index.php/ingenieria/article/view/343943

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