  We develop non-asymptotically justified methods for hypothesis testing about
the $p-$dimensional coefficients $\theta^{*}$ in (possibly nonlinear)
regression models. Given a function $h:\,\mathbb{R}^{p}\mapsto\mathbb{R}^{m}$,
we consider the null hypothesis $H_{0}:\,h(\theta^{*})\in\Omega$ against the
alternative hypothesis $H_{1}:\,h(\theta^{*})\notin\Omega$, where $\Omega$ is a
nonempty closed subset of $\mathbb{R}^{m}$ and $h$ can be nonlinear in
$\theta^{*}$. Our (nonasymptotic) control on the Type I and Type II errors
holds for fixed $n$ and does not rely on well-behaved estimation error or
prediction error; in particular, when the number of restrictions in $H_{0}$ is
large relative to $p-n$, we show it is possible to bypass the sparsity
assumption on $\theta^{*}$ (for both Type I and Type II error control),
regularization on the estimates of $\theta^{*}$, and other inherent challenges
in an inverse problem. We also demonstrate an interesting link between our
framework and Farkas' lemma (in math programming) under uncertainty, which
points to some potential applications of our method outside traditional
hypothesis testing.
