Maximum likelihood estimation from a left-truncated distribution

The traditional approach with experimental raindrop size data has been to use the method of moments (MM) in the fitting procedure to estimate the parameters for the raindrop size distribution function. However, the moment method is known to be biased and can have substantial errors. Some versions of the method of moments may provide estimates of drop-size-distribution (DSD) parameters of sufficient accuracy only when used with very large samples (thousands of drops).

Superior alternative approaches to fitting the observed DSDs are available. Past results of applying the maximum likelihood (ML) method to simulated raindrop samples proved that ML estimators can yield more accurate estimates of DSD parameters than MM estimators when samples include the full range of drop sizes, and this is true whether small or large samples are considered.

However, in the absence of small drops in the samples (typical disdrometer minimum size thresholds are 0.3-0.5 mm) ML shows large bias; this bias does not decrease much with increasing sample size. Therefore, ML estimators will require adaptation to deal with the problem of missing data on small drops. Present work will provide a mathematical description of modified ML estimators for left-truncated distributions and the results of this approach will be tested via computer simulated on known (e.g. gamma) distributions.