# Mean Testing under Truncation beyond Gaussian

arXiv: https://arxiv.org/abs/2605.01335 · HTML: https://mlsanity.com/paper/2605.01335/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 36 references that are in the map.
- First posted on arXiv 2 May 2026.

## Builds on (most cited first, 19 of 19)

- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Recent Advances in Algorithmic High-Dimensional Robust Statistics](https://mlsanity.com/paper/1911.05911/) (2019)
- [Private Mean Estimation of Heavy-Tailed Distributions](https://mlsanity.com/paper/2002.09464/) (2020)
- [Outlier Robust Mean Estimation with Subgaussian Rates via Stability](https://mlsanity.com/paper/2007.15618/) (2020)
- [Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization](https://mlsanity.com/paper/2007.15839/) (2020)
- [Robust subgaussian estimation of a mean vector in nearly linear time](https://mlsanity.com/paper/1906.03058/) (2019)
- [Efficient Statistics, in High Dimensions, from Truncated Samples](https://mlsanity.com/paper/1809.03986/) (2018)
- [Computationally and Statistically Efficient Truncated Regression](https://mlsanity.com/paper/2010.12000/) (2020)
- [Efficient Truncated Statistics with Unknown Truncation](https://mlsanity.com/paper/1908.01034/) (2019)
- [Truncated Linear Regression in High Dimensions](https://mlsanity.com/paper/2007.14539/) (2020)
- [Learning Distributions Generated by One-Layer ReLU Networks](https://mlsanity.com/paper/1909.01812/) (2019)
- [Testing Convex Truncation](https://mlsanity.com/paper/2305.03146/) (2023)
- [The Full Landscape of Robust Mean Testing: Sharp Separations between Oblivious and Adaptive Contamination](https://mlsanity.com/paper/2307.10273/) (2023)
- [What Makes A Good Fisherman? Linear Regression under Self-Selection Bias](https://mlsanity.com/paper/2205.03246/) (2022)
- [Detecting Low-Degree Truncation](https://mlsanity.com/paper/2402.08133/) (2024)
- [Efficient Statistics With Unknown Truncation, Polynomial Time Algorithms, Beyond Gaussians](https://mlsanity.com/paper/2410.01656/) (2024)
- [Beyond Catoni: Sharper Rates for Heavy-Tailed and Robust Mean Estimation](https://mlsanity.com/paper/2311.13010/) (2023)
- [Uniform Mean Estimation for Heavy-Tailed Distributions via Median-of-Means](https://mlsanity.com/paper/2506.14673/) (2025)
- [Private Statistical Estimation via Truncation](https://mlsanity.com/paper/2505.12541/) (2025)

## Abstract (arXiv)

We characterize the fundamental limits of high-dimensional mean testing under arbitrary truncation, where samples are drawn from the conditional distribution P(· mid S) for an unknown truncation set S that may hide up to an varepsilon-fraction of the probability mass. For distributions with p-th directional moments of magnitude at most nuP,p, truncation induces a bias of order O(nuP,pvarepsilon1-1/p). This bias creates a sharp information-theoretic detectability floor: when the signal α falls below this threshold, the null and alternative hypotheses are indistinguishable even with infinite data. Above this floor, we prove that a simple second-order test achieving near-optimal sample complexity n = O!left(frac|SigmaP|(α-4nuP,pvarepsilon1-1/p)2√dright). We further identify a structural escape from this finite-moment bias barrier. Under a directional median regularity assumption, truncation bias improves to linear order O(varepsilon). This reveals an intermediate regime in which estimation requires Theta(d) samples for uniform recovery, while testing recovers the classical Theta(√ d) rate once truncation bias is eliminated. Together, our results provide a unified framework for mean testing under truncation, connecting finite-moment, sub-Gaussian, and median-regular structural regimes.