Minimax Slope Estimation and Support Recovery under Weak Gaussian Selection
Abstract
In sample selection models, identifying a regression slope does not determine how accurately it can be estimated or distinguished from zero. We establish sharp uniform limits for outcome slopes in a fixed-dimensional Gaussian/probit selection model with unknown scale and an unknown covariate density bounded above and away from zero on a cube. We prove a uniform inverse bound between slope differences and observed-law Hellinger distance, allowing every nuisance parameter to differ between the two models. A matching path lies entirely within positive-correlation nonignorable selection. On parameter classes containing a feasible weak-selection neighborhood, the minimax squared slope risk is of order n−2/3. For each prespecified minimum-signal sequence bn, uniform exact support recovery is possible if and only if n1/3bn→∞, with rules allowed to depend on that sequence. The same estimator attains uniform root-n slope accuracy on |γs|≤ M n−1/3 for each fixed finite M≥0, including the exact zero-selection-slope layer. A testing-based construction establishes statistical attainability without estimating the covariate density; computational efficiency is not asserted.
Related
- Minimax Rates for Robust Gaussian Mean Estimation Near the Breakdown Point under Constraints
- Nonparametric hazard rate estimation with associated kernels and minimax bandwidth choice
- Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace–Beltrami Operators on Manifolds
- Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance
- Minimax rates for pointwise CATE estimation: the role of design smoothness
BibTeX
@misc{song2026minimax,
title = {{Minimax Slope Estimation and Support Recovery under Weak Gaussian Selection}},
author = {Zeen Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24749},
eprint = {2610.24749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24749},
}
RIS
TY - UNPB
TI - Minimax Slope Estimation and Support Recovery under Weak Gaussian Selection
AU - Zeen Song
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24749
UR - https://arxiv.org/abs/2610.24749
AB - In sample selection models, identifying a regression slope does not determine how accurately it can be estimated or distinguished from zero. We establish sharp uniform limits for outcome slopes in a fixed-dimensional Gaussian/probit selection model with unknown scale and an unknown covariate density bounded above and away from zero on a cube. We prove a uniform inverse bound between slope differences and observed-law Hellinger distance, allowing every nuisance parameter to differ between the two models. A matching path lies entirely within positive-correlation nonignorable selection. On parameter classes containing a feasible weak-selection neighborhood, the minimax squared slope risk is of order n^−2/3. For each prespecified minimum-signal sequence b_n, uniform exact support recovery is possible if and only if n^1/3b_n→∞, with rules allowed to depend on that sequence. The same estimator attains uniform root-n slope accuracy on |γ_s|≤ M n^−1/3 for each fixed finite M≥0, including the exact zero-selection-slope layer. A testing-based construction establishes statistical attainability without estimating the covariate density; computational efficiency is not asserted.
ER -