Gaussian sequence model with sparse means

Assume the Axiom of Choice as used by Standard normal and normal laws. For integers $d,n\ge 1$, a noise scale $\sigma \ge 0$, and a parameter $\theta \in {\mathbb{R}}^d$, the Gaussian sequence observation is $Y_j={\theta }_j+\frac{\sigma }{\sqrt{n}}{Z}_j,1\le j\le d,$ where $Z_1,\dots ,Z_d$ are independent random variables with law $N(0,1)$. The parameter is $s$-sparse when $0\le s\le d$ is an integer and at most $s$ coordinates of $\theta$ are nonzero. This is an observation model; it does not itself assert a risk bound or an optimal estimator.

[F1] Standard normal and normal laws: under AC, $N(0,1)$ is the probability law with density $(2\pi {)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}2}$. [F2] Independent random elements: independence means independence of the sigma-algebras generated by the random variables; no coordinate independence is inferred unless stated.