Testing and estimation in orthosymmetric Gaussian sequence model

TL;DR

This paper investigates the sample complexity bounds for goodness-of-fit testing in Gaussian sequence models with convex, orthosymmetric parameter spaces, revealing tight bounds and tradeoffs in likelihood-free hypothesis testing.

Contribution

It establishes lower bounds on testing complexity related to estimation complexity for orthosymmetric sets and characterizes LFHT complexity for -bodies, highlighting new sample tradeoffs.

Findings

Goodness-of-fit testing complexity is lower bounded by the square root of estimation complexity.

Tight bounds are achieved when the parameter space is quadratically convex.

New tradeoffs between simulation and observation samples are identified for -bodies.

Abstract

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].