# Finite-Sample Maximum Likelihood Estimation of Location

arXiv: https://arxiv.org/abs/2206.02348 · HTML: https://mlsanity.com/paper/2206.02348/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 14 references that are in the map.
- First posted on arXiv 6 Jun 2022.

## Built on by (most cited first, 7 of 7)

- [Diffusion Posterior Sampling is Computationally Intractable](https://mlsanity.com/paper/2402.12727/) (2024)
- [High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors](https://mlsanity.com/paper/2302.02497/) (2023)
- [Finite-Sample Symmetric Mean Estimation with Fisher Information Rate](https://mlsanity.com/paper/2306.16573/) (2023)
- [Near-Optimal Mean Estimation with Unknown, Heteroskedastic Variances](https://mlsanity.com/paper/2312.02417/) (2023)
- [Attainability of Two-Point Testing Rates for Finite-Sample Location Estimation](https://mlsanity.com/paper/2502.05730/) (2025)
- [Statistical Unlearning of Distributions: A Hypothesis Testing Approach](https://mlsanity.com/paper/2605.16645/) (2026)
- [Near-Efficient and Non-Asymptotic Multiway Inference](https://mlsanity.com/paper/2511.05368/) (2025)

## Abstract (arXiv)

We consider 1-dimensional location estimation, where we estimate a parameter λ from n samples λ + ηi, with each ηi drawn i.i.d. from a known distribution f. For fixed f the maximum-likelihood estimate (MLE) is well-known to be optimal in the limit as n → ∞: it is asymptotically normal with variance matching the Cram'er-Rao lower bound of frac1nI, where I is the Fisher information of f. However, this bound does not hold for finite n, or when f varies with n. We show for arbitrary f and n that one can recover a similar theory based on the Fisher information of a smoothed version of f, where the smoothing radius decays with n.