Open Access
April 2002 On nonparametric tests of positivity/monotonicity/convexity
Anatoli Juditsky, Arkadi Nemirovski
Ann. Statist. 30(2): 498-527 (April 2002). DOI: 10.1214/aos/1021379863


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.


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Anatoli Juditsky. Arkadi Nemirovski. "On nonparametric tests of positivity/monotonicity/convexity." Ann. Statist. 30 (2) 498 - 527, April 2002.


Published: April 2002
First available in Project Euclid: 14 May 2002

zbMATH: 1012.62048
MathSciNet: MR1902897
Digital Object Identifier: 10.1214/aos/1021379863

Primary: 62G08 , 62G10 , 90C25

Keywords: estimation of nonsmooth functionals , minimax risk , Nonparametric test , Tests of convexity

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.30 • No. 2 • April 2002
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