Minimax Slope Estimation and Support Recovery under Weak Gaussian Selection

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 $n^{-2\mathrm{/}3}$. For each prespecified minimum-signal sequence $b_n$, uniform exact support recovery is possible if and only if $n^{1\mathrm{/}3}{b}_n\to \mathrm{\infty }$, with rules allowed to depend on that sequence. The same estimator attains uniform root-$n$ slope accuracy on $\mathrm{\parallel }{\gamma }_s\mathrm{\parallel }\le M{n}^{-1\mathrm{/}3}$ for each fixed finite $M\ge 0$, including the exact zero-selection-slope layer. A testing-based construction establishes statistical attainability without estimating the covariate density; computational efficiency is not asserted.