Part of Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track
Shivam Gupta, Jasper Lee, Eric Price, Paul Valiant
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.