Minimax $L_2$-Separation Rate in Testing the Sobolev-Type Regularity of a function

In this paper we study the problem of testing if an $L_2-$function $f$ belonging to a certain $l_2$-Sobolev-ball $B_t(R)$ of radius $R>0$ with smoothness level $t>0$ indeed exhibits a higher smoothness level $s>t$, that is, belongs to $B_s(R)$. We assume that only a perturbed version of $f$ is available, where the noise is governed by a standard Brownian motion scaled by $\frac{1}{\sqrt{n}}$. More precisely, considering a testing problem of the form $H_0:~f\in B_s(R)~~\mathrm{vs.}~~H_1:~f\in B_t(R),~\inf_{h\in B_s}\Vert f-h\Vert_{L_2}>\rho$ for some $\rho>0$, we approach the task of identifying the smallest value for $\rho$, denoted $\rho^\ast$, enabling the existence of a test $\varphi$ with small error probability in a minimax sense. By deriving lower and upper bounds on $\rho^\ast$, we expose its precise dependence on $n$: $\rho^\ast\sim n^{-\frac{t}{2t+1/2}}.$ As a remarkable aspect of this composite-composite testing problem, it turns out that the rate does not depend on $s$ and is equal to the rate in signal-detection, i.e. the case of a simple null hypothesis.