Empirical-Bayes credible-ball procedure in the inverse Gaussian sequence model

Assume AC. Fix $n\ge 3$, $p\ge 0$, $C\ge 1$, $\gamma \in (0,1)$, an inflation factor $L\ge 0$, and a finite $A_n>0$. Let $({\kappa }_i{)}_{i\ge 1}$ be known nonzero real numbers satisfying $C^{-2}{i}^{-2p}\le {\kappa }_i^2\le C^2{i}^{-2p}\mathrm{.}$ For a true parameter ${\theta }_0\in {\mathrm{\ell }}^2(\mathbb{N},\mathbb{R})$, the observation sequence is $X_i={\kappa }_i{\theta }_{0,i}+n^{-1\mathrm{/}2}{Z}_i,i\ge 1,$ where $(Z_i{)}_{i\ge 1}$ are independent $N(0,1)$ random variables. The corresponding sampling law is denoted $P_{{\theta }_0}^{(n)}$.

For $0<\alpha \le A_n$, let ${\mathrm{\Pi }}_{\alpha }={⨂}_{i\ge 1}N(0,i^{-1-2\alpha })\mathrm{.}$ Its coordinate posterior rule is the product of $N\text{\hspace{0.17em}⁣}({\widehat{\theta }}_{n,\alpha ,i}(x),v_{n,\alpha ,i}),{\widehat{\theta }}_{n,\alpha ,i}(x)=\frac{n{\kappa }_i{x}_i}{i^{1+2\alpha }+n{\kappa }_i^2},v_{n,\alpha ,i}=\frac{1}{i^{1+2\alpha }+n{\kappa }_i^2}\mathrm{.}$ For $\alpha >0$, the marginal log likelihood relative to the reference law ${⨂}_{i\ge 1}N(0,n^{-1})$ is ${\mathrm{\ell }}_n(\alpha ;x)=-\frac{1}{2}{\sum }_{i=1}^{\mathrm{\infty }}[\log \text{\hspace{0.17em}⁣}(1+\frac{n{\kappa }_i^2}{i^{1+2\alpha }})-\frac{n^2{\kappa }_i^2{x}_i^2}{i^{1+2\alpha }+n{\kappa }_i^2}]\mathrm{.}$ For maximization, use this same source score series at every $\alpha \in [0,A_n]$, with the sum taken in increasing coordinate order. Choose a Borel data domain $D_n\subseteq {\mathbb{R}}^{\mathbb{N}}$ such that the series converges to a finite value for every $x\in D_n$ and every $\alpha \in [0,A_n]$, and $\alpha \mapsto {\mathrm{\ell }}_n(\alpha ;x)$ is upper semicontinuous there. For each truth under consideration assume $P_{{\theta }_0}^{(n)}(D_n)=1$. Also assume a Borel-measurable selector $x\mapsto {\widehat{\alpha }}_n(x)$ on $D_n$ is supplied, with ${\widehat{\alpha }}_n(x)\in {\mathrm{arg\hspace{0.17em}max}}_{0\le \alpha \le A_n}{\mathrm{\ell }}_n(\alpha ;x)$. These domain, endpoint-score and selector conditions are explicit procedure hypotheses; no arbitrary score value at $\alpha =0$ is added. At ${\widehat{\alpha }}_n>0$, let $r_{n,\gamma }({\widehat{\alpha }}_n)$ be the $1-\gamma$ quantile of the centered posterior ${\mathrm{\ell }}^2$ norm and use the posterior mean given above. Define the empirical-Bayes set by ${\widehat{C}}_n(L;x)=\{\begin{array}{ll}\{\theta \in {\mathrm{\ell }}^2:\mathrm{\parallel }\theta -{\widehat{\theta }}_{n,{\widehat{\alpha }}_n(x)}(x){\mathrm{\parallel }}_2\le L{r}_{n,\gamma }({\widehat{\alpha }}_n(x))\},& {\widehat{\alpha }}_n(x)>0,\\ {\mathrm{\ell }}^2,& {\widehat{\alpha }}_n(x)=0.\end{array}$ Use the source convention $r_{n,\gamma }(0)=+\mathrm{\infty }$ for the second branch; no posterior probability on ${\mathrm{\ell }}^2$ is asserted at $\alpha =0$. A separate local lemma verifies the coordinate posterior and evidence formulas for $\alpha >0$.

The batch's required type-II definition is stated for a finite-group dominated hierarchy, while this procedure explicitly supplies a sequence model and data-indexed selector. No dominating measure or coverage guarantee from that finite-group setup is imported.