Improved likelihood inference for the power parameter of Tweedie models

The Tweedie family is a class of exponential dispersion models, first introduced by Tweedie (1984), characterized by scale invariance and a variance-to-mean power relationship. Depending on the value of the power parameter, it includes positive and real-valued continuous distributions, discrete distributions, and non-negative mixed distributions, with the normal, Poisson, and gamma families as notable special cases. This dissertation focuses on the Tweedie sub-family with power parameter belonging to (1,2), as this is the non-trivial case most prevalent in practical applications. This model is widely used for non-negative data that have a non-null probability of being exactly zero. Examples include total rainfall, insurance payouts, species abundance in ecological surveys, and time durations. Except for trivial cases, the Tweedie density function does not possess an explicit analytic expression, complicating the use of this family of distributions in statistical modelling. To overcome this limitation, the density is typically evaluated using either highly precise numerical methods, such as the infinite series expansion or the numerical inversion of the characteristic function, or an analytical saddlepoint approximation. A crucial requirement for fitting a generalized linear model with a Tweedie response is the prior estimation of the power parameter, usually obtained via maximisation of the profile likelihood. However, simulation studies conducted in this thesis indicate that the maximum likelihood estimator is highly mean and median biased in finite samples, thus deteriorating the quality of inference and reducing the coverage of confidence intervals: reduction of the power parameter's bias represents an unexplored area in the literature. This project aims to introduce, adapt and compare different methodologies to reduce both the mean and median bias of the maximum likelihood estimator. For mean bias reduction, computationally intensive resampling methods, such as the jackknife and the bootstrap, are compared against higher-order analytical approaches, specifically the explicit estimation of the first-order linear bias term and the implicit score penalty introduced by Firth (1993). For median bias reduction, a bootstrap approach (Cabrera & Watson (1997)) is compared with the implicit score adjustment proposed by Kenne Pagui et al. (2017) and the explicit correction of the MLE (Benussi et al. (2026)). Computing the bias reduction formulas is challenging: the likelihood lacks a closed form, the Tweedie family, with power parameter unknown, does not belong to the exponential dispersion family, and the parameter of interest is not orthogonal to the nuisance parameters' vector. For this reason, analytical corrections will be computed starting from the saddlepoint approximation of the density, which yields quasi-analytical results. The performance of the different techniques will be evaluated through extensive simulation studies, comparing the mean bias, the probability of underestimation, the root mean squared error (RMSE), the standard deviation and the left, right and total coverage of confidence intervals, in different scenarios.