## 1 Introduction  
During the last three decades, mixed Poisson regression models have been applied in various fields of studies, including non-life insurance for modelling overdispersed claim count data. The members of this family of models, which can be constructed based on a mixing distribution which is conjugate to the Poisson distribution, such as the negative binomial and the Poisson-inverse-Gaussian, have been the most popular choices due to the simplicity of their log-likelihood functions, which can be easily maximized using the standard maximum likelihood (ML) estimation approach. See, for example, Refs. [1; 2; 3] for the former and [4; 5; 6] for the latter, among many others. However, it should be noted that the assumption of conjugancy can be very restrictive for constructing a mixed Poisson model that will be able to efficiently capture different levels of overdispersion in real claim count data sets. In particular, as is well known, overdispersion is a direct consequence of unobserved heterogeneity due to systematic effects in the data. For example, in motor insurance, the driving skills, preferences, habits, and driving experience of policyholders, which differ, may lead to extra variation in the claim count data, the degree of which is controlled by the value of the dispersion parameter of the mixed Poisson model. Furthermore, overdispersion can either be caused by a large presence of zeros or a heavy tail in the data. Regarding the latter case, as is well known, the tails of mixed Poisson distributions in the case of continuous mixing distributions are similar to the tails of their mixing distributions (see, for instance, Ref. [7]). Therefore, restricting attention only to mixed Poisson models which are derived based on conjugate to the Poisson mixing densities may result in biased parameter estimates because of their inability to always efficiently model the tail of claim count distribution. This, in turn, may have a profound impact on many tasks which are carried out by the actuaries, such as risk management and pricing of (re-)insurance contracts. Thus, it becomes clear that an important task of actuaries is to be able to design more representative probabilistic models for the number of claims with good prediction accuracy. This procedure depends on the reliability of the statistical method which will be used to construct them. The aim of this article is to present a general class of mixed Poisson regression models with varying dispersion stemming from non-conjugate \(C^{2}\) densities with a continuous first derivative and a continuous second derivative. The class of mixed Poisson models we consider is very wide and the flexibility it provides in (i) the distributional choice for the mixing density and (ii) modelling jointly the mean and dispersion parameters as parametric functions of risk factors allows us to add the required amount of weight to the right tail area of the claim count distribution for accommodating different levels of overdispersion, thus resulting in an improved risk evaluation. At this point, it should be noted that, with the exception of very few articles, such as those by [8; 9; 10], modelling jointly all the parameters of mixed Poisson models in terms of explanatory variables has not been explored in depth. Nevertheless, mean regression models often cannot adequately account for the heteroscedasticity of the claim count distribution or its possible dependence on risk factors. In addition, note that in [9], the exponential family distribution assumption for the univariate response variable is relaxed and replaced by a general distribution family, including distributions based on Box-Cox transformations (such as the Box-Cox t-distribution or the Box-Cox power exponential distribution) and zero adjusted-distributions. However, to the best of our knowledge, this is the first study to consider regression structures on the mean and dispersion parameters of univariate mixed Poisson models that have a probability mass function (pmf) which cannot be written in closed-form expressions.