The Annals of Statistics

Geometrizing Rates of Convergence, II

David L. Donoho and Richard C. Liu

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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.

Article information

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348114

Digital Object Identifier
doi:10.1214/aos/1176348114

Mathematical Reviews number (MathSciNet)
MR1105839

Zentralblatt MATH identifier
0754.62028

JSTOR
links.jstor.org

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

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

Citation

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