14.5 Estimators for Parameters of Drop-size Distribution Functions: Sampling from Gamma Distributions

This work represents a continuation in the investigation of techniques to estimate the parameters for raindrop-size-distribution (DSD) functions. Although moment estimators are often used to estimate these parameters, our previous work (Smith and Kliche, 2005; Smith et al., 2005) showed that the method of moments produces biased results that can lead to misleading extrapolations, and that the resulting functions often do not represent correctly the drop populations. These findings triggered our interest in searching for more robust parameter-fitting techniques. In the present paper we describe some findings of our investigation using Maximum Likelihood (ML) estimators as an alternative to the moment method. The ML method is asymptotically unbiased and seeks to find the parameter values of the assumed gamma size distribution function by maximizing the likelihood function. Monte Carlo simulation of raindrop sampling with various sample sizes was considered in this investigation. In cases where the data encompassed the full range of raindrop sizes, the ML method gave better estimates for the parameters of the gamma DSD function than the moment method. We also investigated the case of poor instrument response at small raindrop sizes by removing the drops smaller than a given threshold from all samples. The MLE parameter values are quite sensitive to missing observations of small drops, and the method of moments often gave superior results in estimating the parameters for the gamma function in these situations.