Distribution Learning in Gaussian Random-Coefficient Regression

We study estimation of an arbitrary coefficient distribution on a fixed Euclidean ball from Gaussian-design linear regression observations. For each fixed dimension, a finite-bit moment estimator achieves expected Wasserstein-1 risk of order 1/log(n), with a matching lower bound for the full observed-pair experiment. Under Huber contamination of fraction eta, the minimax risk is of order max{1/log(n+2), 1/log(e/eta)}. The constructive upper bound also tolerates adversarial replacement of observations. Its proof extends to an unknown additive noise distribution, possibly dependent on the coefficient, under an explicit absolute subgaussian moment envelope and independence of the coefficient-noise pair from the design. Separately, a deterministic inverse modulus combined with the conditional-density theorem of Jiang and Guntuboyina gives the same Wasserstein-1 rate for an exact support-constrained NPMLE under known Gaussian noise. In one dimension, Nguyen's convolution inverse theorem yields the stronger optimal squared Wasserstein-2 rate 1/log(n)^2. The algorithmic bounds are polynomial in sample size for fixed dimension and a fixed noise envelope, not jointly polynomial in dimension and inverse accuracy.