We consider the problem of estimating the distance from an unknown signal, observed in a white-noise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the $L_r$-distance, $1 \leq r < \infty$, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity, monotonicity and convexity of the unknown signal.
We also provide an estimate for the distance to the cone of positive functions for which risk is, by a logarithmic factor, smaller than that of the “plug-in” estimate.
"On nonparametric tests of positivity/monotonicity/convexity." Ann. Statist. 30 (2) 498 - 527, April 2002. https://doi.org/10.1214/aos/1021379863