Geometrizing Rates of Convergence, II

Abstract

Consider estimating a functional $T(F)$ of an unknown distribution $F \in \mathbf{F}$ from data $X_1, \cdots, X_n$ i.i.d. $F$. Let $\omega(\varepsilon)$ denote the modulus of continuity of the functional $T$ over $\mathbf{F}$, computed with respect to Hellinger distance. For well-behaved loss functions $l(t)$, we show that $\inf_{T_n \sup_\mathbf{F}} E_Fl(T_n - T(F))$ is equivalent to $l(\omega(n^{-1/2}))$ to within constants, whenever $T$ is linear and $\mathbf{F}$ 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 $H_0: T(F) \leq t$ and $H_1: T(F) \geq t + \Delta$. 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.