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

Abstract

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.

Modelling and Simulation of Seasonal Rainfall Using the Principle of Maximum Entropy

“…It is known that the introduction of a single correlation parameter allows one to adjust the standard deviation for the sum of the daily totals in a month, so that it matches the observed monthly standard deviation. See, for instance, the correlative coherence analysis proposed by Getz [9], which, incidentally, makes direct use of the Shannon entropy, and the model proposed by Hasan and Dunn [10], which uses a Tweedie distribution. In the overall modelling context, we see it as somewhat logically perverse to select a gamma distribution that will generate realistic daily rainfall depths and then to systematically modify the data generated by it.…”

“…Rosenberg et al [14] constructed a joint density using a Laguerre series to incorporate the correlation between successive months and, hence, to correct the seasonal variance, but the optimal parametric structure of this model is unclear. Hasan and Dunn [10] have recently used a Tweedie distribution to model monthly rainfall. The model combines a Poisson process to generate wet and dry days and a collection of correlated gamma distributions to model daily rainfall depth.…”

Maximum entropy methods for generating simulated rainfall

“…It has been usual to model both short-term and long-term rainfall accumulations at a specific location by a gamma distribution [16,11,3,4]. Some authors [14,5] have, however, observed that simulations in which monthly rainfall totals are modelled as mutually independent gamma random variables generate accumulated bi-monthly, quarterly and yearly totals with much lower variance than the observed accumulations.…”

A Systematic Approach to the Evaluation of RCRA Disposal Facilities Under Future Climate‐Induced Events

“…The exponential dispersion model (EDM) family of distributions includes the response distributions for generalized linear models (GLMs), which have been utilized by several researchers to fit models to input climatological data, such as precipitation (Coe and Stern 1982;Wilks 1999;Chandler 2005;Hasan and Dunn 2011). Recently, there has been a growing interest in developing monthly climate distributions.…”

“…Recently, there has been a growing interest in developing monthly climate distributions. Hasan and Dunn (2011) concluded that not only is this reasonable approach, but also recommend using the EDM family of distributions for this purpose.…”