The Annals of Statistics

Oracle inequalities and adaptive estimation in the convolution structure density model

O. V. Lepski and T. Willer

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We study the problem of nonparametric estimation under $\mathbb{L}_{p}$-loss, $p\in[1,\infty)$, in the framework of the convolution structure density model on $\mathbb{R}^{d}$. This observation scheme is a generalization of two classical statistical models, namely density estimation under direct and indirect observations. The original pointwise selection rule from a family of “kernel-type” estimators is proposed. For the selected estimator, we prove an $\mathbb{L}_{p}$-norm oracle inequality and several of its consequences. Next, the problem of adaptive minimax estimation under $\mathbb{L}_{p}$-loss over the scale of anisotropic Nikol’skii classes is addressed. We fully characterize the behavior of the minimax risk for different relationships between regularity parameters and norm indexes in the definitions of the functional class and of the risk. We prove that the proposed selection rule leads to the construction of an optimally or nearly optimally (up to logarithmic factors) adaptive estimator.

Article information

Ann. Statist., Volume 47, Number 1 (2019), 233-287.

Received: April 2017
Revised: November 2017
First available in Project Euclid: 30 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

Deconvolution model density estimation oracle inequality adaptive estimation kernel estimators $\mathbb{L}_{p}$-risk anisotropic Nikol’skii class


Lepski, O. V.; Willer, T. Oracle inequalities and adaptive estimation in the convolution structure density model. Ann. Statist. 47 (2019), no. 1, 233--287. doi:10.1214/18-AOS1687.

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  • Akakpo, N. (2012). Adaptation to anisotropy and inhomogeneity via dyadic piecewise polynomial selection. Math. Methods Statist. 21 1–28.
  • Birgé, L. (2014). Model selection for density estimation with $\mathbb{L}_{2}$-loss. Probab. Theory Related Fields 158 533–574.
  • Butucea, C. and Tsybakov, A. B. (2008). Sharp optimality in density deconvolution with dominating bias. I, II. Theory Probab. Appl. 52 111–128, 237–249.
  • Comte, F. and Lacour, C. (2013). Anisotropic adaptive kernel deconvolution. Ann. Inst. Henri Poincaré Probab. Stat. 49 569–609.
  • Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006). Penalized contrast estimator for adaptive density deconvolution. Canad. J. Statist. 34 431–452.
  • de Guzmán, M. (1975). Differentiation of Integrals in $R^{n}$. Lecture Notes in Mathematics 481. Springer, Berlin.
  • Devroye, L. and Lugosi, G. (1997). Nonasymptotic universal smoothing factors, kernel complexity and Yatracos classes. Ann. Statist. 25 2626–2637.
  • Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508–539.
  • Duval, C. (2017). A note on a fixed-point method for deconvolution. Statistics 51 347–362.
  • Efroĭmovich, S. Yu. (1986). Nonparametric estimation of a density of unknown smoothness. Theory Probab. Appl. 30 557–568.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
  • Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600–610.
  • Fan, J. and Koo, J.-Y. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734–747.
  • Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley, New York.
  • Gach, F., Nickl, R. and Spokoiny, V. (2013). Spatially adaptive density estimation by localised Haar projections. Ann. Inst. Henri Poincaré Probab. Stat. 49 900–914.
  • Giné, E. and Nickl, R. (2009). An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation. Probab. Theory Related Fields 143 569–596.
  • Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. Ann. Statist. 39 1608–1632.
  • Goldenshluger, A. and Lepski, O. (2014). On adaptive minimax density estimation on $R^{d}$. Probab. Theory Related Fields 159 479–543.
  • Golubev, G. K. (1992). Non-parametric estimation of smooth probability densities. Probl. Inf. Transm. 1 52–62.
  • Grafakos, L. (2008). Classical Fourier Analysis, 2nd ed. Graduate Texts in Mathematics 249. Springer, New York.
  • Hall, P. and Meister, A. (2007). A ridge-parameter approach to deconvolution. Ann. Statist. 35 1535–1558.
  • Hasminskii, R. and Ibragimov, I. (1990). On density estimation in the view of Kolmogorov’s ideas in approximation theory. Ann. Statist. 18 999–1010.
  • Hesse, C. H. (1995). Deconvolving a density from partially contaminated observations. J. Multivariate Anal. 55 246–260.
  • Juditsky, A. and Lambert-Lacroix, S. (2004). On minimax density estimation on $\mathbb{R}$. Bernoulli 10 187–220.
  • Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising. Probab. Theory Related Fields 121 137–170.
  • Kerkyacharian, G., Pham Ngoc, T. M. and Picard, D. (2011). Localized spherical deconvolution. Ann. Statist. 39 1042–1068.
  • Lepski, O. (2013). Multivariate density estimation under sup-norm loss: Oracle approach, adaptation and independence structure. Ann. Statist. 41 1005–1034.
  • Lepski, O. (2015). Adaptive estimation over anisotropic functional classes via oracle approach. Ann. Statist. 43 1178–1242.
  • Lepski, O. V. (2018). A new approach to estimator selection. Bernoulli 24. To appear. Available at arXiv:1603.03934v1.
  • Lepski, O. V. and Willer, T. (2017). Lower bounds in the convolution structure density model. Bernoulli 23 884–926.
  • Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 201–231.
  • Masry, E. (1993). Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stochastic Process. Appl. 47 53–74.
  • Meister, A. (2009). Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics 193. Springer, Berlin.
  • Nikol’skii, S. M. (1977). Priblizhenie Funktsii Mnogikh Peremennykh i Teoremy Vlozheniya [Approximation of Functions of Several Variables and Imbedding Theorems], 2nd ed., revised and supplemented. Nauka, Moscow (in Russian).
  • Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033–2053.
  • Rebelles, G. (2016). Structural adaptive deconvolution under $\mathbb{L}_{p}$-losses. Math. Methods Statist. 25 26–53.
  • Reynaud-Bouret, P., Rivoirard, V. and Tuleau-Malot, C. (2011). Adaptive density estimation: A curse of support? J. Statist. Plann. Inference 141 115–139.
  • Rigollet, P. (2006). Adaptive density estimation using the blockwise Stein method. Bernoulli 12 351–370.
  • Rigollet, Ph. and Tsybakov, A. B. (2007). Linear and convex aggregation of density estimators. Math. Methods Statist. 16 260–280.
  • Samarov, A. and Tsybakov, A. (2007). Aggregation of density estimators and dimension reduction. In Advances in Statistical Modeling and Inference. Ser. Biostat. 3 233–251. World Sci. Publ., Hackensack, NJ.
  • Stefanski, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229–235.
  • Stefanski, L. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169–184.
  • Yuan, M. and Chen, J. (2002). Deconvolving multidimensional density from partially contaminated observations. J. Statist. Plann. Inference 104 147–160.