Estimating moments of subjectively assessed distributions

Moment-matching discrete distributions were developed by Miller and Rice as a method to translate continuous probability distributions into discrete distributions for use in decision and risk analysis. Using gaussian quadrature, they showed that an n-point discrete distribution can be constructed that exactly matches the first 2n − 1 moments of the underlying distribution. These moment-matching discrete distributions offer several theoretical advantages over the typical discrete approximations as shown in Smith, but they also pose practical problems. In particular, how does the analyst estimate the moments given only the subjective assessments of the continuous probability distribution? Smith suggests that the moments can be estimated by fitting a distribution to the assessments. This research note shows that the quality of the moment estimates cannot be judged solely by how close the fitted distribution is to the true distribution. Examples are used to show that the relative errors in higher order moment estimates can be greater than 100%, even though the cumulative distribution function is estimated within a Kolmogorov–Smirnov distance less than 1%.