Fitting a distribution by the first two moments (partial and complete)

Given a sample of observations from an unknown population, a common practice to derive distributional representation for the given data is to fit a four-parameter distribution via matching of the first four moments. However, third and fourth sample moments are notorious for their large standard errors, which require sample sizes that in a typical industrial setting are rarely available. In this paper we propose an alternative approach that employs only the first two moments (partial and complete) to fit a certain four-parameter distribution to the given sample data. The fitted distribution is a mixture of two components, where each is a linear transformation of a symmetrically distributed standardized variable. Separate transformations are used for each half of the distribution. Estimation of the parameters is carried out by matching of the mean, the variance, and the first and second partial moments. This fitting procedure is shown to be approximately a least squares solution, that provides good-estimates for the fractiles of the approximated distribution. Moreover, the linear transformations may provide mathematically manageable solutions to stochastic optimization problems (like inventory problems) that would otherwise require complex solution procedures. Some numerical examples and a simulation study attest to the effectiveness of the new approach when sample data are scarce.