Testing and estimation in orthosymmetric Gaussian sequence model

Abstract

We study the Gaussian sequence model, i.e. $X\sim N(\hat{\mathbf{I}}¸,I_{\infty })$, where $\hat{\mathbf{I}}¸\in \hat{I}“\subset {\ell }_2$ 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 $\hat{I}“$ is orthosymmetric. This lower bound is tight when $\hat{I}“$ 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].

Title changed. Old title was "Gaussian Sequence Model: Sample Complexities of Testing, Estimation and LFHT"