Minimax detection risk and separation rate

Assume Countable Choice and use the Gaussian sequence inverse model with singular values $s_j>0$, inverse-noise weights ${\sigma }_j=s_j^{-1}$, and parameter coordinates $\theta =({\theta }_j)\in {\mathrm{\ell }}^2$. Let $a=(a_j)$ be a positive nondecreasing sequence with $a_j\to \mathrm{\infty }$, and let $\sigma =({\sigma }_j)$ be a positive nondecreasing sequence with ${\sigma }_j\to \mathrm{\infty }$. For $\epsilon >0$, put ${\eta }_j={\theta }_j\mathrm{/}{\sigma }_j$ and observe $Y_j=Z_j\mathrm{/}{\sigma }_j={\eta }_j+\epsilon {\xi }_j$, where $Z$ is the normalized observation in the Gaussian sequence inverse model and the ${\xi }_j$ are independent standard normal random variables. For $0<q\le 2$ and $r>0$, define ${\mathrm{\Theta }}_q(r)=\{\eta \in {\mathbb{R}}^{{\mathbb{N}}_{>0}}:({\sigma }_j{\eta }_j{)}_{j\ge 1}\in {\mathrm{\ell }}^2,{\sum }_{j=1}^{\mathrm{\infty }}\mathrm{\mid }{a}_j{\sigma }_j{\eta }_j{\mathrm{\mid }}^q\le 1,{\sum }_{j=1}^{\mathrm{\infty }}{\sigma }_j^2{\eta }_j^2\ge r^2\}\mathrm{.}$ Test $H_0:\eta =0$ against $H_1:\eta \in {\mathrm{\Theta }}_q(r)$ by a product-Borel measurable randomized test $\psi :{\mathbb{R}}^{{\mathbb{N}}_{>0}}\to [0,1]$, where $\psi (y)$ is the rejection probability. Its type-I error and maximal type-II error are ${\alpha }_{\epsilon }(\psi )={\mathbb{E}}_{\epsilon ,0}\psi ,{\beta }_{\epsilon }({\mathrm{\Theta }}_q(r),\psi )={\sup }_{\eta \in {\mathrm{\Theta }}_q(r)}{\mathbb{E}}_{\epsilon ,\eta }(1-\psi )\mathrm{.}$ When ${\mathrm{\Theta }}_q(r)=\mathrm{\varnothing }$, define ${\beta }_{\epsilon }({\mathrm{\Theta }}_q(r),\psi )={\beta }_{\epsilon }({\mathrm{\Theta }}_q(r),\alpha )={\gamma }_{\epsilon }({\mathrm{\Theta }}_q(r))=0$, the convention that no type-II error occurs when no alternative is possible. For $\alpha \in (0,1)$ define ${\beta }_{\epsilon }({\mathrm{\Theta }}_q(r),\alpha )={\inf }_{\psi :{\alpha }_{\epsilon }(\psi )\le \alpha }{\beta }_{\epsilon }({\mathrm{\Theta }}_q(r),\psi ),{\gamma }_{\epsilon }({\mathrm{\Theta }}_q(r))={\inf }_{\psi }\{{\alpha }_{\epsilon }(\psi )+{\beta }_{\epsilon }({\mathrm{\Theta }}_q(r),\psi )\}\mathrm{.}$ A positive family $r_{\epsilon }^{\ast }\to 0$ is a minimax separation rate if ${\gamma }_{\epsilon }({\mathrm{\Theta }}_q(r_{\epsilon }))\to 1$ and ${\beta }_{\epsilon }({\mathrm{\Theta }}_q(r_{\epsilon }),\alpha )\to 1-\alpha$ for every $\alpha \in (0,1)$ whenever $r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to 0$, while ${\gamma }_{\epsilon }({\mathrm{\Theta }}_q(r_{\epsilon }))\to 0$ and ${\beta }_{\epsilon }({\mathrm{\Theta }}_q(r_{\epsilon }),\alpha )\to 0$ for every $\alpha \in (0,1)$ whenever $r_{\epsilon }\mathrm{/}{r}_{\epsilon }^{\ast }\to \mathrm{\infty }$.

The Gaussian sequence inverse model and the coordinate law $Z_j={\theta }_j+\epsilon s_j^{-1}{\xi }_j$

Countable Choice, which is assumed by the upstream sequence model

The null-first type-I/type-II orientation of the testing function

Ingster, Sapatinas and Suslina, Minimax signal detection in ill-posed inverse problems, §2, PDF pp. 4–6

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §6.2.1 and §6.2.3, PDF pp. 489–494 and 506–508