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

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Abstract

The gamma family of probability densities has recently been used to model raindrop size. However, the traditional approach of using the method of moments to estimate the gamma distribution parameters is known to be biased and can have substantial errors. A recently developed approach combining moment information and a weighted least-squares analysis generally produces substantially better results. Other procedures 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 for the case in which the full range of drop sizes are observed and for the case in which small drop sizes fail to be observed because of the inability of disdrometers to record observations below a threshold.

The foregoing comments on maximum likelihood 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 many disdrometers; this requires some modification of the maximum likelihood procedure. We also allow for the possibility of drop sizes below a threshold, or above another threshold, not being observed. Maximum likelihood performance in this case is investigated through simulation of volume sampling from gamma distributions with known parameters. We compare the performance of the maximum likelihood estimates with those of method of moments (only a truncated-data version is viable) and the recently developed weighted least-squares procedure, and also apply the three estimation procedures to some experimental data. Since the experimental data are surface data, we indicate how drop fall velocity may be incorporated to obtain parameters for the volume distributions from the surface data.

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