# On the approximation of sum of lognormal for correlated variates and implementation

## TL;DR

This paper compares three methods for approximating the sum of correlated lognormal variables and finds that the Wilkinson method performs best across various scenarios.

## Contribution

The study provides new empirical evidence that the Wilkinson method outperforms others for correlated lognormal sums.

## Key findings

- Wilkinson approximation has the lowest Type I error rates across all tested correlation structures and sample sizes.
- Results were validated using real-world datasets from engineering and finance, confirming Wilkinson's superior performance.
- The findings contradict prior telecommunications literature that favored the Schwartz & Yeh method.

## Abstract

In probabilistic modeling across engineering, finance, and telecommunications, sums of lognormal random variables frequently occur, yet no closed-form expression exists for their distribution. This study systematically evaluates three approximation methods—Wilkinson (W), Schwartz & Yeh (SY), and Inverse (I)—for correlated lognormal variates across varying sample sizes and correlation structures. Using Monte Carlo simulations with 5, 15, 25, and 30 samples and correlation coefficients of 0.3, 0.6, and 0.9, we compared Type I error rates through Anderson-Darling goodness-of-fit tests. Our findings demonstrate that the Wilkinson approximation consistently outperforms the other methods for correlated variates, exhibiting the lowest Type I error rates across all tested scenarios. This contradicts some previous findings in telecommunications literature where SY was preferred. We validated these results using real-world datasets from engineering (fatigue life of ball bearings) and finance (stock price correlations), confirming the Wilkinson approximation’s superior performance through probability density function comparisons. This research provides practical guidance for selecting appropriate approximation methods when modeling correlated lognormal sums in diverse applications.