Maximum Likelihood Estimation for Bivariate Joint Distribution Recovery from Max-Aggregated Data

In modern communication systems, to conserve transmission energy, the collected data are often max-aggregated. This aggregation involves observing only the features with relatively larger values in each observed sample. Recovering the joint distribution from such systematically missing data is of great interest for numerous downstream applications, including network optimization and user localization. This paper is motivated by the aforementioned industrial problem and investigates the recoverability and estimation of the joint distribution in the bivariate case. We consider various parametric model assumptions (uniform, Gaussian, mixture of Gaussians) and derive corresponding loss functions based on the maximum likelihood (ML) principle. The effectiveness of the proposed method is demonstrated through simulations. We also discuss its potential application in future wireless communication networks.