Abstract: 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 $n \to \infty$, 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.