The Annals of Statistics

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

Cristina Butucea

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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

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

First available in Project Euclid: 7 November 2007

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Zentralblatt MATH identifier

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

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


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.

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  • Bickel, P. J. and Ritov, Y. (1988). Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhyā Ser. A 50 381–393.
  • Birgé, L. and Massart, P. (1995). Estimation of integral functionals of a density. Ann. Statist. 23 11–29.
  • Butucea, C. (2004). Asymptotic normality of the integrated square error of a density estimator in the convolution model. SORT 28 9–25.
  • Butacea, C. (2004). Goodness-of-fit testing and quadratic functional estimation from indirect observations. Long version with Appendix. Available at
  • Butucea, C. and Matias, C. (2005). Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model. Bernoulli 11 309–340.
  • Butucea, C. and Tsybakov, A. B. (2007). Sharp optimality for density deconvolution with dominating bias. I, II. Theory Probab. Appl. 51. To appear.
  • Carroll, R. J., Ruppert, D. and Stefanski, L. A. (1995). Measurement Error in Nonlinear Models. Chapman and Hall, London.
  • Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006). Penalized contrast estimator for density deconvolution. Canad. J. Statist. 34 431–452.
  • Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249–267.
  • Efromovich, S. and Low, M. (1996). On optimal adaptive estimation of a quadratic functional. Ann. Statist. 24 1106–1125.
  • Ermakov, M. S. (1994). Minimax nonparametric testing of hypotheses on a distribution density. Theory Probab. Appl. 39 396–416.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
  • Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273–1294.
  • Fromont, M. and Laurent, B. (2006). Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 680–720.
  • Gayraud, G. and Pouet, C. (2005). Adaptive minimax testing in the discrete regression scheme. Probab. Theory Related Fields 133 531–558.
  • Hall, P. and Marron, J. S. (1987). Estimation of integrated squared density derivatives. Statist. Probab. Lett. 6 109–115.
  • Ibragimov, I. A. and Khas'minskii, R. Z. (1991). Asymptotically normal families of distributions and efficient estimation. Ann. Statist. 19 1681–1724.
  • Ingster, Yu. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III. Math. Methods Statist. 2 85–114, 171–189, 249–268.,
  • Kerkyacharian, G. and Picard, D. (1996). Estimating nonquadratic functionals of a density using Haar wavelets. Ann. Statist. 24 485–507.
  • Laurent, B. (1996). Efficient estimation of integral functionals of a density. Ann. Statist. 24 659–681.
  • Lepski, O. V. and Levit, B. Y. (1998). Adaptive minimax estimation of infinitely differentiable functions. Math. Methods Statist. 7 123–156.
  • Lepski, O. V. and Tsybakov, A. B. (2000). Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Related Fields 117 17–48.
  • Levit, B. Ya. (1978). Asymptotically efficient estimation of nonlinear functionals. Problems Inform. Transmission 14 204–209.
  • Lukacs, E. (1970). Characteristic Functions, 2nd ed. Hafner, New York.
  • Nemirovski, A. (2000). Topics in non-parametric statistics. Lectures on Probability Theory and Statistics. Lecture Notes in Math 1738 85–277. Springer, Berlin.
  • Pouet, C. (1999). On testing nonparametric hypotheses for analytic regression functions in Gaussian noise. Math. Methods Statist. 8 536–549.
  • Richardson, S. and Green, P. J. (1997). On Bayesian analysis of mixtures with an unknown number of components (with discussion). J. Roy. Statist. Soc. Ser. B 59 731–792.
  • Roeder, K. and Wasserman, L. (1997). Practical Bayesian density estimation using mixtures of normals. J. Amer. Statist. Assoc. 92 894–902.
  • Speed, T., ed. (2003). Statistical Analysis of Gene Expression Microarray Data. Chapman and Hall/CRC, Boca Raton, FL.
  • Spokoiny, V. G. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477–2498.
  • Stephens, M. (2000). Bayesian analysis of mixture models with an unknown number of components–-an alternative to reversible jump methods. Ann. Statist. 28 40–74.
  • Tribouley, K. (2000). Adaptive estimation of integrated functionals. Math. Methods Statist. 9 19–38.