Two Tweedie distributions that are near-optimal for modelling monthly rainfall in Australia

摘要

Statistical models for total monthly rainfall used for forecasting, risk management and agricultural simulations are usually based on gamma distributions and variations. In this study, we examine a family of distributions (called the Tweedie family of distributions) to determine if the choice of the gamma distribution is optimal within the family. We restrict ourselves to the exponential family of distributions as they are the response distributions used for generalized linear models (GLMs), which has numerous advantages. Further, we restrict ourselves to distributions where the variance is proportional to some power of the mean, as these distributions also have desirable properties. Under these restrictions, an infinite number of distributions exist for modelling positive continuous data and include the gamma distribution as a special case. Results show that for positive monthly rainfall totals in the data history for a particular station, monthly rainfall is optimally or near-optimally modelled using the gamma distribution by varying the parameters of the gamma distribution; using different distributions for each month cannot improve on this approach. In addition, under the same model restrictions, monthly rainfall totals that include zeros are also well modelled by the same family of distributions. Hence monthly rainfall can be suitably modelled using one of two Tweedie distributions depending on whether exact zeros appear in the rainfall history. We propose a slight variation of the gamma distribution for use in practice. This model fits the data almost as well as the gamma distribution but admits the possibility that future months may have zero rainfall.