Minimax-Optimal Location Estimation

Location estimation is one of the most basic questions in parametric statistics. Suppose we have a known distribution density f, and we get n i.i.d. samples from f(x - mu) for some unknown shift mu. The task is to estimate mu to high accuracy with high probability. The maximum likelihood estimator (MLE) is known to be asymptotically optimal as n -> infinity, but what is possible for finite n? In this paper, we give two location estimators that are optimal under different criteria: 1) an estimator that has minimax-optimal estimation error subject to succeeding with probability 1 - delta and 2) a confidence interval estimator which, subject to its output interval containing mu with probability at least 1 - delta, has the minimum expected squared interval width among all shift-invariant estimators. The latter construction can be generalized to minimizing the expectation of any loss function on the interval width.