Gaussian Sequence Model: Sample Complexities of Testing, Estimation and LFHT

We study the Gaussian sequence model, i.e. $X \sim N(\mathbf{\theta}, I_\infty)$, where $\mathbf{\theta} \in \Gamma \subset \ell_2$ is assumed to be convex and compact. We show that goodness-of-fit testing sample complexity is lower bounded by the square-root of the estimation complexity, whenever $\Gamma$ is orthosymmetric. We show that the lower bound is tight when $\Gamma$ is also quadratically convex, thus significantly extending validity of the testing-estimation relationship from [GP24]. Using similar methods, we also completely characterize likelihood-free hypothesis testing (LFHT) complexity for $\ell_p$-bodies, discovering new types of tradeoff between the numbers of simulation and observation samples.