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June, 1991 Geometrizing Rates of Convergence, II
David L. Donoho, Richard C. Liu
Ann. Statist. 19(2): 633-667 (June, 1991). DOI: 10.1214/aos/1176348114


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.


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David L. Donoho. Richard C. Liu. "Geometrizing Rates of Convergence, II." Ann. Statist. 19 (2) 633 - 667, June, 1991.


Published: June, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0754.62028
MathSciNet: MR1105839
Digital Object Identifier: 10.1214/aos/1176348114

Primary: 62G20
Secondary: 62F35 , 62G05

Keywords: Density estimation , estimating the mode , estimating the rate of tail decay , Hellinger distance , minimax tests , modulus of continuity , monotone likelihood ratio , robust nonparametric regression

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • June, 1991
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