A joint sparse representation (JSR) method has shown superior performance for the classification of hyperspectral images (HSIs). However, it is prone to be affected by outliers in the HSI spatial neighborhood. In order to improve the robustness of JSR, we propose a maximum likelihood estimation (MLE)-based JSR (MLEJSR) model, which replaces the traditional quadratic loss function with an MLE-like estimator for measuring the joint approximation error. The MLE-like estimator is actually a function of coding residuals. Given some priors on the coding residuals, the MLEJSR model can be easily converted to an iteratively reweighted JSR problem. Choosing a reasonable weight function, the effect of inhomogeneous neighboring pixels or outliers can be dramatically reduced. We provide a theoretical analysis of MLEJSR from the viewpoint of recovery error and evaluate its empirical performance on three public hyperspectral data sets. Both the theoretical and experimental results demonstrate the effectiveness of our proposed MLEJSR method, especially in the case of large noise.