This article considers parameter estimation, goodness of fit, likelihood ratio and score tests, and model selection by Akaike information criterion for the inverse trinomial (IT) distribution, a classical one-dimensional random walk distribution. The IT distribution has a cubic variance function of the mean and is a generalization of the negative binomial distribution. Basic distributional properties and expressions for the probability mass function, recurrence formula, moments, and score functions are also presented.

This paper considers the construction of a family of discrete distributions with the flexibility to cater for under-, equi- and over-dispersion in count data using a finite mixture model based on standard distributions. We are motivated to introduce this family because its simple finite mixture structure adds flexibility and facilitates application and use in analysis. The family of distributions is exemplified using a mixture of negative binomial and shifted negative binomial distributions. Some basic and probabilistic properties are derived. We perform hypothesis testing for equi-dispersion and simulation studies of their power and consider parameter estimation via maximum likelihood and probability-generating-function-based methods. The utility of the distributions is illustrated via their application to real biological data sets exhibiting under-, equi- and over-dispersion. It is shown that the distribution fits better than the well-known generalized Poisson and COM–Poisson distributions for handling under-, equi- and over-dispersion in count data.

In the transportation services industry, the proper assessment of insurance claim count distribution is an important step to determine insurance premiums based on policyholders’ risk profiles. Risk factors are identified through regression analysis. In this paper, the inverse trinomial distribution is proposed as a count data model for insurance claims characterised by having long tails and a high index of dispersion. Two regression models are developed to identify associated risk factors. Other popular models, such as the negative binomial and COM-Poisson, are fitted and compared to information criteria. The risk profiles of policyholders are determined based on the selected model. To illustrate the application of the inverse trinomial regression models, the ausprivautolong dataset of automobile claims in Australia has been fitted with identification of risk factors.

Modeling bivariate count data with complex dispersion and dependence structures remains a significant challenge in statistical data analysis. This article introduces two new bivariate count distributions derived from the generalized shifted inverse trinomial distribution. The proposed models, denoted by BGIT-I and BGIT-II, are constructed using convolution and trivariate reduction methods. They provide flexible joint frameworks for modeling correlated count data while accommodating different marginal dispersion patterns. BGIT-I allows negative, near-zero, and positive dependence, whereas BGIT-II induces non-negative dependence through a common component. The proposed models have simple, tractable probability generating functions, which facilitate the derivation of probabilistic properties and motivate a probability-generating-function-based estimation approach alongside maximum-likelihood estimation. The finite-sample performance of the estimators is further examined through a Monte Carlo simulation study. The practical utility of the proposed models is illustrated using two real bivariate count data sets involving shunter accidents and patient counts in critical care and emergency room settings. The results show that the proposed BGIT models provide competitive alternatives for modeling bivariate count data with different dispersion and dependence characteristics.

In this paper, we propose a modified mean-variance portfolio selection procedure using the proximal gradient method. This novel selection procedure has the advantage of encouraging sparsity in the entire portfolio usingl₀−norm while requiring only O(n) storage. The proximal gradient method (PSG) introduces a multiple damping gradient (MDG) method to reduce the computation time, with Lipschitz constant serving as the step size, and iterative thresholding (IHT) as the proximal method to induce the sparsity of the portfolio. To find the optimal portfolio, an efficient algorithm is developed. The performance of the proposed selection procedure is illustrated by its application to real-life data set. It is shown that the proposed portfolio performs consistently better in comparison with other methods.