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In, we have recently proposed a general approach for approximating the power sum of Log-Normal Random Variables (RVs) by using the Pearson system of distributions. Therein, we have also highlighted the main advantages of using Pearson approximation instead of the usual Log-Normal one, and compared the proposed method with other approaches available in the open technical literature. However, despite being very accurate, the proposed method may be, in some circumstances, computational demanding since a non-linear least-squares problem needs to be solved numerically to get an accurate approximation. Motivated by the above consideration, the aim of this Letter is to provide an alternative approach for computing the parameters of the approximating Pearson distribution. The proposed solution is based on the Method of Moments (MoMs) in the logarithmic domain. In particular, by using some known properties of the Laplace transform, we will show that the MGF of the Log-Normal power sum in the logarithmic domain (denoted as log-MGF) can be obtained from the Mellin transform of the MGF of the Log-Normal power sum in the linear domain. From the estimated log-MGF, we will then compute the desired log-moments required for Pearson approximation. Numerical results will be also shown in order to substantiate the accuracy of the proposed method.