= Global argmin may escape a compact convergence set
{title2=$a(\theta)=\theta^2/(1+\theta^4)$}
<Uniform convergence in probability> of empirical criteria on a compact set proves consistency for minimizers in that set; it does not confine a minimizer over a larger domain. A bounded identifiable regression function can approach its true value again at infinity. For median regression, $a(\theta)=\theta^2/(1+\theta^4)$ has unique zero at zero, but a small positive fitted <sample median> $c$ has a second solution $\theta^2=(1+\sqrt{1-4c^2})/(2c)$. This solution diverges as $c\downarrow0$, although uniform convergence on any fixed compact interval holds. Constraining the estimator, or proving suitable global separation and localization, repairs the argument.
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