Mildly ill-posed analytic q=2 detection rate

Let $m$ be the largest active index. Then $A{e}^{2\alpha m}<1\le A{e}^{2\alpha (m+1)}$, so $t=A{e}^{2\alpha m}\in [e^{-2\alpha },1)$. For $j=m-\mathrm{\ell }$ with $0\le \mathrm{\ell }<m$, the profile factor is ${\rho }_{\mathrm{\ell }}=1-t{e}^{-2\alpha \mathrm{\ell }}$. For $\mathrm{\ell }\ge 1$, $1-e^{-2\alpha }\le {\rho }_{\mathrm{\ell }}\le 1$. Hence the terms with $1\le \mathrm{\ell }\le m\mathrm{/}2$ give a lower bound for $J_1$ and $J_0$, while the full sums are bounded above by ${\sum }_{j\le m}{j}^{4\beta }\le m^{4\beta +1}$. Thus $J_1\asymp J_0\asymp m^{4\beta +1}$. In $J_2$, the summand at $\mathrm{\ell }$ is $t{e}^{-2\alpha \mathrm{\ell }}(m-\mathrm{\ell }{)}^{4\beta }{\rho }_{\mathrm{\ell }}$. Its sum is at most $m^{4\beta }{\sum }_{\mathrm{\ell }\ge 0}{e}^{-2\alpha \mathrm{\ell }}\le C_{\alpha }{m}^{4\beta }$, and the $\mathrm{\ell }=1$ term is at least $c_{\alpha ,\beta }{m}^{4\beta }$ for $m\ge 2$. Therefore $J_2\asymp m^{4\beta }$. Formula [F1] gives $r^2\asymp e^{-2\alpha m}m,u_{\epsilon }^2\asymp {\epsilon }^{-4}{r}^4{m}^{-(4\beta +1)}\mathrm{.}$ Taking logarithms of the first relation shows $m\asymp \log (1\mathrm{/}r)$.

Set $L=\log {\epsilon }^{-1}$ and $r_{\epsilon }^{\ast }=\epsilon L^{\beta +1\mathrm{/}4}$. At this radius, $\log (1\mathrm{/}{r}_{\epsilon }^{\ast })\sim L$, so [F1] and step 1.1 give $u_{\epsilon }^2\asymp 1$. If $r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to 0$, then $\log (1\mathrm{/}{r}_{\epsilon })\ge \log (1\mathrm{/}{r}_{\epsilon }^{\ast })\sim L$ and $u_{\epsilon }^2\asymp (r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }{)}^4{\left(\frac{\log (1\mathrm{/}{r}_{\epsilon }^{\ast })}{\log (1\mathrm{/}{r}_{\epsilon })}\right)}^{4\beta +1}\to 0.$ If $r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to \mathrm{\infty }$ and $r_{\epsilon }\to 0$, the logarithm ratio is at least one, so $u_{\epsilon }\to \mathrm{\infty }$. If the radii do not tend to zero, use ${\bar{r}}_{\epsilon }=\sqrt{r_{\epsilon }{r}_{\epsilon }^{\ast }}$. Nonempty alternatives satisfy $r_{\epsilon }^2\le a_1^{-2}$ because ${\sum }_j{\sigma }_j^2{\eta }_j^2\le a_1^{-2}{\sum }_j{a}_j^2{\sigma }_j^2{\eta }_j^2\le a_1^{-2}$. Hence ${\bar{r}}_{\epsilon }\to 0$, ${\bar{r}}_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to \mathrm{\infty }$, and ${\mathrm{\Theta }}_2(r_{\epsilon })\subseteq {\mathrm{\Theta }}_2({\bar{r}}_{\epsilon })$; consistent tests for the larger class work on the original class. The two risk implications in [F1] prove the claimed rate.