Modeling Loss Exchange Ratios as Inverse Gaussian Variates: Implications

Abstract

This paper outlines a method for estimating and comparing the Loss Exchange Ratio (LER) output of computer combat simulations, and develops methods to establish a priori the number of simulation runs required to detect a change in the parameters of the simulation of a given size. The Loss Exchange Ratio (LER) is a widely used and widely accepted summary statistic for a simulation run involving force-on-force combat models. The LER is surprisingly variable - multiple runs of the same scenario produce a large range of LER. We assert here that these loss exchange ratios are skewed stochastic random variables, and that they are well modeled by the inverse gaussian (IG) distribution. We discuss technical reasons for preferring the inverse gaussian model over other distributions, particularly the log-normal distribution. Adopting this IG stochastic model allows us to develop explicit statistical methods for estimating the parameters of this distribution, using its known sampling distributions. We also inherit precise statistical tests for hypothesis testing. Finally, we are able to determine a priori the number of simulation runs necessary to detect a change in the distribution of a given size. This is a particularly valuable ability, given the increased reliance of the Army on these simulation models to make procurement and doctrinal decisions. We discuss how these simulation tests fit into the larger scheme of procurement and doctrine decisions We illustrate with data sets from both the JANUS and CASTFOREM simulations. In particular, we find that the use of the IG model allows us to make more powerful conclusions about the data. In our discussion of the inverse gaussian distribution, we illustrate the versatility of this lesser known member of the exponential family for modeling positive skewed data and provide a primer on its properties. We conclude that the IG is a good model for describing the variability of LER with useful estimation and testing properties, and recommend its consideration when modeling LER. We sketch two other promising areas for research which follow from the use of this model.