The Annals of Statistics

Goodness-of-fit testing and quadratic functional estimation from indirect observations

Cristina Butucea

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Abstract

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.

Article information

Source
Ann. Statist., Volume 35, Number 5 (2007), 1907-1930.

Dates
First available in Project Euclid: 7 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1194461716

Digital Object Identifier
doi:10.1214/009053607000000118

Mathematical Reviews number (MathSciNet)
MR2363957

Zentralblatt MATH identifier
1126.62028

Subjects
Primary: 62F12: Asymptotic properties of estimators 62G05: Estimation 62G10: Hypothesis testing 62G20: Asymptotic properties

Keywords
Asymptotic efficiency convolution model exact constant in nonparametric tests goodness-of-fit tests infinitely differentiable functions quadratic functional estimation minimax tests Sobolev classes

Citation

Butucea, Cristina. Goodness-of-fit testing and quadratic functional estimation from indirect observations. Ann. Statist. 35 (2007), no. 5, 1907--1930. doi:10.1214/009053607000000118. https://projecteuclid.org/euclid.aos/1194461716


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