In this paper, a novel Bayesian estimator for the minimum bounding axis-aligned rectangle of a point set based on noisy measurements is derived. Each given measurement stems from an unknown point and is corrupted with additive Gaussian noise. Extreme value theory is applied in order to derive a linear measurement equation for the problem. The new estimator is applied to the problem of group target and extended object tracking. Instead of estimating each single group member or point feature explicitly, the basic idea is to track a summarizing shape, namely the minimum bounding rectangle, of the group. Simulation results demonstrate the feasibility of the estimator.