Geometrizing Rates of Convergence, II

Consider estimating a functional $T(F)$ of an unknown distribution F ∈ F from data X₁, ⋯, Xₙ i.i.d. F. Let ω(ε) denote the modulus of continuity of the functional T over F, computed with respect to Hellinger distance. For well-behaved loss functions $l(t)$, we show that infTₙ F EFl(Tₙ - T(F)) is equivalent to l(ω(n-1/2)) to within constants, whenever T is linear and F is convex. The same conclusion holds in three nonlinear cases: estimating the rate of decay of a density, estimating the mode and robust nonparametric regression. We study the difficulty of testing between the composite, infinite dimensional hypotheses H₀: T(F) ≤ t and H₁: T(F) ≥ t + Δ. Our results hold, in the cases studied, because the difficulty of the full infinite-dimensional composite testing problem is comparable to the difficulty of the hardest simple two-point testing subproblem.