Closed‐form parameter estimates for a truncated gamma distribution

Gamma distributions are used by atmospheric scientists to model the distribution of raindrop diameters. However, the recording instruments (known as disdrometers) are unable to record the sizes of the smallest drops, so that the recorded distribution (which consists of group frequencies) is truncated. Although the method of maximum likelihood can be used to estimate the parameters of such a truncated and grouped distribution, this method is not customarily used by practitioners. Their preference has been for the very simple method of moments. Many different combinations of moments have been suggested, with a preference for the use of higher moments both because of their physical relevance (to the volume of water, the attenuation of radio signals or radar reflectivity) and because of the low‐end truncation. However, the difficulty in accurately estimating higher moments leads to these estimates often being biased and invariably having high variance. Recognising the attraction of a computationally simple procedure, we introduce a method that uses a combination of moment relationships and weighted linear regression to give explicit formulae for parameter estimates. We compare the performance of these estimates with both maximum likelihood and moment estimates for untruncated, singly truncated and doubly truncated gamma distributions. We show that our simple closed‐form estimates compare favourably with the optimal maximum likelihood estimates.