A Novel Highly Accurate Log Skew Normal Approximation Method to Lognormal Sum Distributions

Abstract

Sums of lognormal random variables occur in many important problems in wireless communications. However, the lognormal sum distribution is known to have no close-form and is difficult to compute numerically. Several approximation methods have already been proposed to approximate the lognormal sum distribution. However, these approximation methods all have their drawbacks: some widely used approximation methods are not very accurate at the lower region, some other approximation methods require the CDF curve from Monte Carlo simulation first. In this paper, we propose a novel approximation method, namely the Log Skew Normal (LSN) approximation, to model and approximate the sum of M lognormal distributed random variables. The proposed LSN approximation method has very high accuracy in most of the region, especially in the lower region. Furthermore, this approximation method does not require the CDF curve from Monte Carlo simulation first. The closed-form probability density function (PDF) of the resulting LSN random variable is presented and its parameters are derived from those of the M individual lognormal random variables by using an moment matching technique. Simulation results on the cumulative distribution function (CDF) of sum of M lognormal random variables in different conditions are used as reference curves to compare various approximation techniques. LSN approximation is found to provide better accuracy over a wide CDF range over other approximation methods.