Goodness-of-fit testing and quadratic functional estimation from indirect observations. (English) Zbl 1126.62028

Summary: We consider the convolution model where i.i.d. random variables \(X_i\) having unknown density \(f\) are observed with additive i.i.d. noise, independent of the \(X_i\)’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 \(\int f^{2}\) 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.


62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
Full Text: DOI arXiv Euclid


[1] Bickel, P. J. and Ritov, Y. (1988). Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhyā Ser. A 50 381–393. · Zbl 0676.62037
[2] Birgé, L. and Massart, P. (1995). Estimation of integral functionals of a density. Ann. Statist. 23 11–29. · Zbl 0848.62022
[3] Butucea, C. (2004). Asymptotic normality of the integrated square error of a density estimator in the convolution model. SORT 28 9–25. · Zbl 1274.62229
[4] Butacea, C. (2004). Goodness-of-fit testing and quadratic functional estimation from indirect observations. Long version with Appendix. Available at arxiv.org/abs/math/0612361.
[5] 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. · Zbl 1063.62044
[6] Butucea, C. and Tsybakov, A. B. (2007). Sharp optimality for density deconvolution with dominating bias. I, II. Theory Probab. Appl. 51 . · Zbl 1142.62017
[7] Carroll, R. J., Ruppert, D. and Stefanski, L. A. (1995). Measurement Error in Nonlinear Models . Chapman and Hall, London. · Zbl 0853.62048
[8] Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006). Penalized contrast estimator for density deconvolution. Canad. J. Statist. 34 431–452. · Zbl 1104.62033
[9] Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249–267. · Zbl 1429.62125
[10] Efromovich, S. and Low, M. (1996). On optimal adaptive estimation of a quadratic functional. Ann. Statist. 24 1106–1125. · Zbl 0865.62024
[11] Ermakov, M. S. (1994). Minimax nonparametric testing of hypotheses on a distribution density. Theory Probab. Appl. 39 396–416. · Zbl 0855.62032
[12] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272. · Zbl 0729.62033
[13] Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273–1294. · Zbl 0729.62076
[14] Fromont, M. and Laurent, B. (2006). Adaptive goodness-of-fit tests in a density model. Ann. Statist. 34 680–720. · Zbl 1096.62040
[15] Gayraud, G. and Pouet, C. (2005). Adaptive minimax testing in the discrete regression scheme. Probab. Theory Related Fields 133 531–558. · Zbl 1075.62029
[16] Hall, P. and Marron, J. S. (1987). Estimation of integrated squared density derivatives. Statist. Probab. Lett. 6 109–115. · Zbl 0628.62029
[17] Ibragimov, I. A. and Khas’minskii, R. Z. (1991). Asymptotically normal families of distributions and efficient estimation. Ann. Statist. 19 1681–1724. · Zbl 0760.62043
[18] Ingster, Yu. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III. Math. Methods Statist. 2 85–114, 171–189, 249–268., Mathematical Reviews (MathSciNet): Mathematical Reviews (MathSciNet): MR1259685 · Zbl 0798.62059
[19] Kerkyacharian, G. and Picard, D. (1996). Estimating nonquadratic functionals of a density using Haar wavelets. Ann. Statist. 24 485–507. · Zbl 0860.62033
[20] Laurent, B. (1996). Efficient estimation of integral functionals of a density. Ann. Statist. 24 659–681. · Zbl 0859.62038
[21] Lepski, O. V. and Levit, B. Y. (1998). Adaptive minimax estimation of infinitely differentiable functions. Math. Methods Statist. 7 123–156. · Zbl 1103.62332
[22] 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. · Zbl 0971.62022
[23] Levit, B. Ya. (1978). Asymptotically efficient estimation of nonlinear functionals. Problems Inform. Transmission 14 204–209. · Zbl 0422.62034
[24] Lukacs, E. (1970). Characteristic Functions , 2nd ed. Hafner, New York. · Zbl 0201.20404
[25] Nemirovski, A. (2000). Topics in non-parametric statistics. Lectures on Probability Theory and Statistics . Lecture Notes in Math 1738 85–277. Springer, Berlin. · Zbl 0998.62033
[26] Pouet, C. (1999). On testing nonparametric hypotheses for analytic regression functions in Gaussian noise. Math. Methods Statist. 8 536–549. · Zbl 1103.62344
[27] 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. JSTOR: · Zbl 0891.62020
[28] Roeder, K. and Wasserman, L. (1997). Practical Bayesian density estimation using mixtures of normals. J. Amer. Statist. Assoc. 92 894–902. JSTOR: · Zbl 0889.62021
[29] Speed, T., ed. (2003). Statistical Analysis of Gene Expression Microarray Data. Chapman and Hall/CRC, Boca Raton, FL. · Zbl 1108.62331
[30] Spokoiny, V. G. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477–2498. · Zbl 0898.62056
[31] 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. · Zbl 1106.62316
[32] Tribouley, K. (2000). Adaptive estimation of integrated functionals. Math. Methods Statist. 9 19–38. · Zbl 1006.62037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.