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October 2007 Goodness-of-fit testing and quadratic functional estimation from indirect observations
Cristina Butucea
Ann. Statist. 35(5): 1907-1930 (October 2007). DOI: 10.1214/009053607000000118


We consider the convolution model where i.i.d. random variables Xi having unknown density f are observed with additive i.i.d. noise, independent of the X’s. We assume that the density f belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically.

We consider the problem of goodness-of-fit testing in the convolution model. We prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional ∫ f2 based on indirect observations. When the unknown density is smoother enough than the noise density, we prove that this estimator is n−1/2 consistent, asymptotically normal and efficient (for the variance we compute). Otherwise, we give nonparametric upper bounds for the risk of the same estimator.

We give an approach unifying the proof of nonparametric minimax lower bounds for both problems. We establish them for Sobolev densities and for supersmooth densities less smooth than exponential noise. In the two setups we obtain exact testing constants associated with the asymptotic minimax rates.


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Cristina Butucea. "Goodness-of-fit testing and quadratic functional estimation from indirect observations." Ann. Statist. 35 (5) 1907 - 1930, October 2007.


Published: October 2007
First available in Project Euclid: 7 November 2007

zbMATH: 1126.62028
MathSciNet: MR2363957
Digital Object Identifier: 10.1214/009053607000000118

Primary: 62F12 , 62G05 , 62G10 , 62G20

Keywords: Asymptotic efficiency , Convolution model , exact constant in nonparametric tests , Goodness-of-fit tests , Infinitely differentiable functions , minimax tests , Quadratic functional estimation , Sobolev classes

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.35 • No. 5 • October 2007
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