Marked Point Process Models of Raindrop-Size Distributions

Large Sample Comparison of Parameter Estimates in Gamma Raindrop Distributions

Raindrop size distributions have been characterized through the gamma family. Over the years, quite a few estimates of these gamma parameters have been proposed. The natural question for the practitioner, then, is what estimation procedure should be used. We provide guidance in answering this question when a large sample size (>2000 drops) of accurately measured drops is available. Seven estimation procedures from the literature: five method of moments procedures, maximum likelihood, and a pseudo maximum likelihood procedure, were examined. We show that the two maximum likelihood procedures provide the best precision (lowest variance) in estimating the gamma parameters. Method of moments procedures involving higher-order moments, on the other hand, give rise to poor precision (high variance) in estimating these parameters. A technique called the delta method assisted in our comparison of these various estimation procedures.

The Bias in Moment Estimators for Parameters of Drop Size Distribution Functions: Sampling from Exponential Distributions

The moment estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions are biased. Consequently, the fitted functions often do not represent well either the raindrop samples or the underlying populations from which the samples were taken. Monte Carlo simulations of the process of sampling from a known exponential DSD, followed by the application of a variety of moment estimators, demonstrate this bias. Skewness in the sampling distributions of the DSD moments is the root cause of this bias, and this skewness increases with the order of the moment. As a result, the bias is stronger when higher-order moments are used in the procedures. Correlations of the sample moments with the size of the largest drop in a sample (Dmax) lead to correlations of the estimated parameters with Dmax, and, in turn, to spurious correlations between the parameters. These things can lead to erroneous inferences about characteristics of the raindrop populations that are being sampled. The bias, and the correlations, diminish as the sample size increases, so that with large samples the moment estimators may become sufficiently accurate for many purposes.