Mid-quantile mixed graphical models with an application to mass public shootings in the U.S

Mass public shootings in the U.S. have become a major public health hazard, impacting the safety and well-being of individuals and communities. Motivated by this pressing issue, we propose a mid-quantile mixed graphical model for investigating the intricacies of inter- and infra-domain relationships of this complex phenomenon, where conditional relations between discrete and continuous variables are modelled without stringent distributional assumptions using Parzen's definition of mid-quantile. To retrieve the graph structure and recover only the most relevant connections, we consider the neighbourhood selection approach in which conditional mid-quantiles of each variable in the network are modelled as a sparse function of all others. We propose a two-step procedure to estimate the graph where, in the first step, conditional mid-probabilities are obtained semi-parametrically and, in the second step, the model parameters are estimated by solving an implicit equation with a LASSO penalty.