High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors

In location estimation, we are given n samples from a known distribution f shifted by an unknown translation λ, and want to estimate λ as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cramér-Rao bound of error N (0, 1 nI ), where I is the Fisher information of f . However, the n required for convergence depends on f , and may be arbitrarily large. We build on the theory using smoothed estimators to bound the error for finite n in terms of I r , the Fisher information of the r-smoothed distribution. As n → ∞, r → 0 at an explicit rate and this converges to the Cramér-Rao bound. We (1) improve the prior work for 1-dimensional f to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest. Gupta et al. (2022) addressed this question in the special case of 1 dimension. They showed that, while the MLE can have bad finite-sample performance, it is possible to improve the behavior by a simple adaptation: add Gaussian noise of some appropriately chosen radius r, where r decreases with the number of samples, to both the samples and model before performing MLE. Accordingly, the theoretical guarantees for the smoothed MLE replaces the Fisher information of f with the Fisher information of the smoothed distribution f r , also called the Recent work by