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. This lower bound is tight when $\Gamma$ is also quadratically convex (as shown by [Donoho et al. 1990, Neykov 2023]). 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, compared to the case of ellipsoids (p = 2) studied in [Gerber and Polyanskiy 2024].

翻译：我们研究高斯序列模型，即 $X \sim N(\mathbf{\theta}, I_\infty)$，其中假设 $\mathbf{\theta} \in \Gamma \subset \ell_2$ 是凸且紧的。我们证明，只要 $\Gamma$ 是正交对称的，拟合优度检验的样本复杂度下界由估计复杂度的平方根给出。当 $\Gamma$ 同时也是二次凸时，该下界是紧的（如 [Donoho et al. 1990, Neykov 2023] 所示）。我们还完整刻画了 $\ell_p$-球体上的无似然假设检验（LFHT）复杂度，与 [Gerber and Polyanskiy 2024] 中研究的椭球（p = 2）情形相比，发现了仿真样本与观测样本数量之间新的权衡类型。