Abstract In this article, we propose and study the class of multivariate log‐normal/independent distributions and linear regression models based on this class. The class of multivariate log‐normal/independent distributions is very attractive for robust statistical modeling because it includes several heavy‐tailed distributions suitable for modeling correlated multivariate positive data that are skewed and possibly heavy‐tailed. Besides, expectation‐maximization (EM)‐type algorithms can be easily implemented for maximum likelihood estimation. We model the relationship between quantiles of the response variables and a set of explanatory variables, compute the maximum likelihood estimates of parameters through EM‐type algorithms, and evaluate the model fitting based on Mahalanobis‐type distances. The satisfactory performance of the quantile estimation is verified by simulation studies. An application to newborn data is presented and discussed.