• Bernoulli
  • Volume 23, Number 2 (2017), 884-926.

Lower bounds in the convolution structure density model

O.V. Lepski and T. Willer

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The aim of the paper is to establish asymptotic lower bounds for the minimax risk in two generalized forms of the density deconvolution problem. The observation consists of an independent and identically distributed (i.i.d.) sample of $n$ random vectors in $\mathbb{R}^{d}$. Their common probability distribution function $\mathfrak{p}$ can be written as $\mathfrak{p}=(1-\alpha)f+\alpha[f\star g]$, where $f$ is the unknown function to be estimated, $g$ is a known function, $\alpha$ is a known proportion, and $\star$ denotes the convolution product. The bounds on the risk are established in a very general minimax setting and for moderately ill posed convolutions. Our results show notably that neither the ill-posedness nor the proportion $\alpha$ play any role in the bounds whenever $\alpha\in[0,1)$, and that a particular inconsistency zone appears for some values of the parameters. Moreover, we introduce an additional boundedness condition on $f$ and we show that the inconsistency zone then disappears.

Article information

Bernoulli, Volume 23, Number 2 (2017), 884-926.

Received: January 2015
Revised: May 2015
First available in Project Euclid: 4 February 2017

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Zentralblatt MATH identifier

$\mathbb{L}_{p}$-risk adaptive estimation density estimation generalized deconvolution model minimax rates Nikol’skii spaces


Lepski, O.V.; Willer, T. Lower bounds in the convolution structure density model. Bernoulli 23 (2017), no. 2, 884--926. doi:10.3150/15-BEJ763.

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  • [1] Butucea, C. (2004). Deconvolution of supersmooth densities with smooth noise. Canad. J. Statist. 32 181–192.
  • [2] Butucea, C. and Tsybakov, A.B. (2007). Sharp optimality in density deconvolution with dominating bias. I. Theory Probab. Appl. 52 24–39.
  • [3] Butucea, C. and Tsybakov, A.B. (2007). Sharp optimality in density deconvolution with dominating bias. II. Theory Probab. Appl. 52 237–249.
  • [4] Carroll, R.J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
  • [5] Comte, F. and Lacour, C. (2013). Anisotropic adaptive kernel deconvolution. Ann. Inst. Henri Poincaré Probab. Stat. 49 569–609.
  • [6] Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006). Penalized contrast estimator for adaptive density deconvolution. Canad. J. Statist. 34 431–452.
  • [7] Devroye, L. (1989). The double kernel method in density estimation. Ann. Inst. Henri Poincaré Probab. Stat. 25 533–580.
  • [8] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508–539.
  • [9] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
  • [10] Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600–610.
  • [11] Fan, J. and Koo, J.-Y. (2002). Wavelet deconvolution. IEEE Trans. Inform. Theory 48 734–747.
  • [12] Goldenshluger, A. (1999). On pointwise adaptive nonparametric deconvolution. Bernoulli 5 907–925.
  • [13] Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. Ann. Statist. 39 1608–1632.
  • [14] Goldenshluger, A. and Lepski, O. (2014). On adaptive minimax density estimation on $R^{d}$. Probab. Theory Related Fields 159 479–543.
  • [15] Hall, P. and Meister, A. (2007). A ridge-parameter approach to deconvolution. Ann. Statist. 35 1535–1558.
  • [16] Johnstone, I.M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 547–573.
  • [17] Johnstone, I.M. and Raimondo, M. (2004). Periodic boxcar deconvolution and Diophantine approximation. Ann. Statist. 32 1781–1804.
  • [18] Juditsky, A. and Lambert-Lacroix, S. (2004). On minimax density estimation on $\mathbb{R}$. Bernoulli 10 187–220.
  • [19] Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising. Probab. Theory Related Fields 121 137–170.
  • [20] Lepski, O. (2013). Multivariate density estimation under sup-norm loss: Oracle approach, adaptation and independence structure. Ann. Statist. 41 1005–1034.
  • [21] Lepski, O. (2015). Adaptive estimation over anisotropic functional classes via oracle approach. Ann. Statist. 43 1178–1242.
  • [22] Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 201–231.
  • [23] Masry, E. (1993). Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stochastic Process. Appl. 47 53–74.
  • [24] Meister, A. (2009). Deconvolution Problems in Nonparametric Statistics. Lecture Notes in Statistics 193. Berlin: Springer.
  • [25] Nikol’skiĭ, S.M. (1977). Priblizhenie Funktsii Mnogikh Peremennykh i Teoremy Vlozheniya, 2nd ed. Moscow: Nauka.
  • [26] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033–2053.
  • [27] Rebelles, G. (2015). Structural adaptive deconvolution under $L_{p}$-losses. Preprint. Available at arXiv:1504.06246v1.
  • [28] Stefanski, L. and Carroll, R.J. (1990). Deconvoluting kernel density estimators. Statistics 21 169–184.
  • [29] Stefanski, L.A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229–235.
  • [30] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. New York: Springer.
  • [31] von Bahr, B. and Esseen, C.-G. (1965). Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq2$. Ann. Math. Statist 36 299–303.
  • [32] Willer, T. (2006). Deconvolution in white noise with a random blurring effect. Preprint LPMA.