Tweedie exponential dispersion models: theory, properties and inference

Tweedie models are a class of exponential dispersion models (EDMs) characterised by scale invariance and a variance-to-mean power relationship. Depending on the value of the power parameter, the Tweedie family includes positive continuous distributions, real-valued continuous distributions, discrete distributions and non-negative mixed distributions, which have a positive probability of taking a value of zero. Several well-known probability distributions, such as the normal, Poisson, gamma and inverse Gaussian, are included in this family. Tweedie EDMs have been extensively used to model total amounts of physical phenomena (such as total rainfall and total insurance payouts), time duration and variables with highly skewed distributions, consistently outperforming alternative models. However, except for the aforementioned notable cases, their density function does not possess an explicit analytic expression, complicating the use of this family of distributions in statistical modelling. In Chapter 4, maximum likelihood inference is illustrated with a particular focus on estimating the power parameter via profile likelihood. This latter chapter includes the implementation of a function in R that computes an estimate and a confidence interval for the Tweedie parameter, analogous to the ‘tweedie.profile’ function from the package ‘tweedie’ (Dunn, 2017). In Chapter 5, simulation studies are conducted to examine the properties of likelihood inference about the power parameter, as computed by the aforementioned R function, for various values of the power parameter and two different sample sizes.