Approximating the Sum of Correlated Lognormal or, Lognormal-Rice Random Variables

Abstract

A simple and novel method is presented to approximate by the lognormal distribution the probability density function of the sum of correlated lognormal random variables. The method is also shown to work well for approximating the distribution of the sum of lognormal-Rice or Suzuki random variables by the lognormal distribution. The method is based on matching a low-order Gauss-Hermite approximation of the moment-generating function of the sum of random variables with that of a lognormal distribution at a small number of points. Compared with methods available in the literature such as the Fenton-Wilkinson method, Schwartz-Yeh method, and their extensions, the proposed method provides the parametric flexibility to address the inevitable trade-off that needs to be made in approximating different regions of the probability distribution function. ICC 2006This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Abstract-A simple and novel method is presented to approximate by the lognormal distribution the probability density function of the sum of correlated lognormal random variables. The method is also shown to work well for approximating the distribution of the sum of lognormal-Rice or Suzuki random variables by the lognormal distribution. The method is based on matching a low-order Gauss-Hermite approximation of the moment-generating function of the sum of random variables with that of a lognormal distribution at a small number of points. Compared with methods available in the literature such as the Fenton-Wilkinson method, Schwartz-Yeh method, and their extensions, the proposed method provides the parametric flexibility to address the inevitable trade-off that needs to be made in approximating different regions of the probability distribution function.

Fitting the Modified-Power-Lognormal to the Sum of Independent Lognormals Distribution

“…Now there exist several numerical methods that could perform better than our MPLN method in many cases. We should, however, make the following comments: 1) Several numerical methods approximate the SLN by another lognormal [1], [12], [19]. From Figures 1-3 and from literature, it is becoming evident that in many cases of interest this cannot possibly be accurate, since the SLN distribution is very non-linear on lognormal paper.…”

“…This is substantially simpler to understand and implement than other numerical methods [11], [12], [17], [19]- [21], [23], [24]. The first moment of the MPLN distribution can be written as…”

“…While the execution of these algorithms on a computer may be quasi-instantaneous, they still require significant human investment to understand and implement the method. Method [12] is perhaps not very complicated. However our method is much simpler to understand and can be implemented quickly with only a simple Riemann sum.…”

Stochastic Volatility Effects on Correlated Log-Normal Random Variables

“…L 1 contains the mixed partial derivatives due to the 2 )) of . Then, solutions of (10) are given by the form…”

“…In addition, analytic approximation formulas retain qualitative model information and preserve an explicit dependence of the results on the underlying parameters. Many approximate methods can be categorized by their scope into three classes: generally correlated [7][8][9][10], independent [11][12][13][14], and independent and identically distributed [15,16]. The probability density function of the correlated log-normal especially is carried out to propose various approximations of the sum distribution.…”