Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 9 / b / iii / Solution

The joint maximum-likelihood estimators are functions of the observations selecting a maximizing parameter pair:The estimates after observation maximize the same likelihood function with $X_i=x_i$. Maximization may be nonunique, as for an even sample size, or may fail to yield a finite admissible pair, as in the one-observation case below. The definition does not turn a boundary supremum into an attained maximum.