Quantile-based graphical models for continuous and discrete variables

In this paper we develop a mixed graphical model for identifying conditional independence relations between continuous and discrete variables in a quantile framework using Parzen’s definition of mid-quantile. To recover the graph structure and induce sparsity, we consider the neighborhood selection approach in which conditional mid-quantiles of each variable in the network are modeled as a sparse function of all others. Building on previous work, we propose a two-step estimation procedure where, in the first step, conditional mid-probabilities are obtained and, in the second step, the model parameters are estimated by solving an implicit equation with a LASSO penalty. The empirical application investigates the relationship between depression and inflammation on a sample of individuals from the National Health and Nutrition Examination Survey 2017-2020.