Maximum likelihood estimation of gamma parameters for coarsely-binned and left-truncated raindrop size data

The gamma family of densities, which includes the exponential family as a special case, has recently been used to model raindrop size data. The traditional approach of using method of moments to estimate the gamma distribution parameters, however, is known to be biased and can have substantial errors. Methods superior to the method of moments approach include maximum likelihood. In particular, maximum likelihood estimates have been shown to outperform method of moments estimators both in the case in which the full range of drop sizes are observed as well as the case in which small drop sizes fail to be observed because of the inability of disdrometers to record observations below a threshold. The comments above concern the situation in which drop sizes are measured on a continuous scale. In this work we consider drop sizes from gamma distributions which are classified into broad size bins, as would be the case with data obtained from an impact disdrometer; we do allow for the possibility of drop sizes below a threshold not being observed. Maximum likelihood performance in this case is investigated through simulation from gamma distributions with known parameters. The simulation process, which relies in part on numerical optimization as the maximum likelihood estimates are not expressible in closed-form, is conducted using the R statistical package (http://www.r-project.org/).