Exact characterization of the optimal slack factor γ* for robust Hellinger testing

Determine the exact value of the optimal slack factor γ*, defined as the smallest γ > 1 such that for every pair of distributions {p1, p2} over a common domain, there exists a test that, given finitely many i.i.d. samples from an arbitrary target distribution p, can decide between γ H^2(p, p1) ≤ H^2(p, p2) and H^2(p, p2) ≥ γ H^2(p, p1) with error bounded by a fixed constant δ < 1/2, thereby returning the distribution in {p1, p2} that is closer to p in squared Hellinger distance.

Background

The paper studies robust hypothesis testing under model misspecification, aiming to output the distribution among {p1, p2} that is closer to the true distribution p in squared Hellinger distance. Because p may be nearly equidistant to p1 and p2, the formulation introduces a slack factor γ > 1 to ensure separability, and defines γ-robustness via consistent decision with bounded error.

Prior work established upper bounds on γ* for Hellinger distance via tests of Suresh and Baraud, with γ* ≤ (√2 + 1)/(√2 − 1). This paper proves a lower bound γ* ≥ √2/(√2 − 1) and shows tightness under disjoint supports, leaving a gap between bounds in the general case. The precise value of γ* remains undetermined and is explicitly stated as an open problem.