Multivariate Negative Binomial Weighted Lindley Generalized Linear Model: Methodological Innovations and Applications in Traffic Safety

The concept of a multivariate distribution is essential in statistics, allowing the simultaneous analysis of related variables. In transportation safety, such models are effective for studying crash data across multiple categories, improving predictions and evaluations of safety measures. This paper extends the negative binomial weighted Lindley (NB-WLindley) distribution, known for handling highly dispersed or sparse data, into a multivariate framework. The proposed multivariate NB-WLindley generalized linear model treats each crash category as a random variable dependent on other categories within a joint framework while preserving the marginal distributional properties. It is hierarchically defined as a mixture of NB and multivariate weighted Lindley distributions and incorporates a dependence structure to explain correlations among categories. Applications to two crash datasets show that the multivariate NB-WLindley model can simultaneously capture different crash types and severities, identifying dependencies that univariate models cannot. The study also develops a random parameters version of the model to address unobserved heterogeneity, which consistently outperforms the fixed-parameter version and yields stronger predictive performance. Overall, this work demonstrates that the proposed multivariate model offers a more flexible and accurate approach to crash analysis, enhancing the ability to capture variability and interdependence across crash categories. It provides a practical tool for improving safety assessments and supporting data-driven decision-making in transportation safety research.