Convex Bodies and Maximum Likelihood Sets

We consider points sampled from a uniform distribution on a convex body in high dimensional real space with unknown location. In this case, the maximum likelihood estimator set is a convex body containing the true location parameter, and hence has a volume and diameter. We estimate these quantities, in terms of dimension and number of samples, by introducing upper and lower bounds. These bounds are different depending on the geometry of the convex body. We arrive at our results by employing algebraic, probabilistic and statistical techniques.