The Annals of Statistics

Geometrizing Rates of Convergence, II

David L. Donoho and Richard C. Liu

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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.

Article information

Ann. Statist., Volume 19, Number 2 (1991), 633-667.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G20: Asymptotic properties
Secondary: 62G05: Estimation 62F35: Robustness and adaptive procedures

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


Donoho, David L.; Liu, Richard C. Geometrizing Rates of Convergence, II. Ann. Statist. 19 (1991), no. 2, 633--667. doi:10.1214/aos/1176348114.

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See also

  • Part III: David L. Donoho, Richard C. Liu. Geometrizing Rates of Convergence, III. Ann. Statist., Volume 19, Number 2 (1991), 668--701.