The Annals of Statistics

On nonparametric tests of positivity/monotonicity/convexity

Anatoli Juditsky and Arkadi Nemirovski

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Abstract

We consider the problem of estimating the distance from an unknown signal, observed in a white-noise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown function belongs to a Hölder class, the risk of estimating the $L_r$-distance, $1 \leq r < \infty$, from the signal to a cone is essentially the same (up to a logarithmic factor) as that of estimating the signal itself. The same risk bounds hold for the test of positivity, monotonicity and convexity of the unknown signal.

We also provide an estimate for the distance to the cone of positive functions for which risk is, by a logarithmic factor, smaller than that of the “plug-in” estimate.

Article information

Source
Ann. Statist., Volume 30, Number 2 (2002), 498-527.

Dates
First available in Project Euclid: 14 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1021379863

Digital Object Identifier
doi:10.1214/aos/1021379863

Mathematical Reviews number (MathSciNet)
MR1902897

Zentralblatt MATH identifier
1012.62048

Subjects
Primary: 62G10: Hypothesis testing 62G08: Nonparametric regression 90C25: Convex programming

Keywords
Tests of convexity nonparametric test estimation of nonsmooth functionals minimax risk

Citation

Juditsky, Anatoli; Nemirovski, Arkadi. On nonparametric tests of positivity/monotonicity/convexity. Ann. Statist. 30 (2002), no. 2, 498--527. doi:10.1214/aos/1021379863. https://projecteuclid.org/euclid.aos/1021379863


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