The bias In moment estimators for parameters of drop-size distribution functions: sampling from gamma distributions

This paper complements an earlier one (Smith & Kliche, 2005), showing that the moment estimators frequently used to estimate parameters for drop-size-distribution (DSD) functions being “fitted” to observed raindrop size distributions are biased. 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 known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case), followed by 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; this skewness decreases as the shape parameter of the population gamma DSD increases, but increases with the order of the moment. As a result, the bias is stronger when higher-order moments are used in the procedures. The sample moments are correlated with the size of the largest drop in a sample (Dmax), but the correlations are weaker than in the case of sampling from an exponential DSD and the correlations between the estimated parameters and Dmax noted for that case disappear. However, spurious correlations between the parameters are even stronger (though non-linear) than for the exponential DSD case. These things can lead to erroneous inferences about characteristics of the raindrop populations 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.

The maximum likelihood estimators (MLEs) suggested by some earlier authors are also being applied to these simulations. For the estimates of the gamma shape parameter (which tends to be badly overestimated by the moment methods), the bias is minimal and the scatter in the MLE estimates is much smaller than that for the moment estimators.