Adaptive estimation of linear functionals in the convolution model and applications

C. Butucea and F. Comte

Full-text: Open access


We consider the model $Z_i=X_i+\varepsilon_i$, for i.i.d. $X_i$’s and $\varepsilon_i$’s and independent sequences $(X_i)_{i∈ℕ}$ and $(\varepsilon_i)_{i∈ℕ}$. The density $f_\varepsilon$ of $\varepsilon_1$ is assumed to be known, whereas the one of $X_1$, denoted by $g$, is unknown. Our aim is to estimate linear functionals of $g, 〈ψ, g〉$ for a known function $ψ$. We propose a general estimator of $〈ψ, g〉$ and study the rate of convergence of its quadratic risk as a function of the smoothness of $g, f_\varepsilon$ and $ψ$. Different contexts with dependent data, such as stochastic volatility and AutoRegressive Conditionally Heteroskedastic models, are also considered. An estimator which is adaptive to the smoothness of unknown $g$ is then proposed, following a method studied by Laurent et al. (Preprint (2006)) in the Gaussian white noise model. We give upper bounds and asymptotic lower bounds of the quadratic risk of this estimator. The results are applied to adaptive pointwise deconvolution, in which context losses in the adaptive rates are shown to be optimal in the minimax sense. They are also applied in the context of the stochastic volatility model.

Article information

Bernoulli, Volume 15, Number 1 (2009), 69-98.

First available in Project Euclid: 3 February 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

adaptive density estimation ARCH models deconvolution linear functionals model selection penalized contrast stochastic volatility model


Butucea, C.; Comte, F. Adaptive estimation of linear functionals in the convolution model and applications. Bernoulli 15 (2009), no. 1, 69--98. doi:10.3150/08-BEJ146.

Export citation


  • [1] Artiles Martínez, L.M. (2001). Adaptive minimax estimation in classes of smooth functions. Ph.D. thesis. Utrecht University, Netherlands.
  • [2] Artiles Martínez, L.M. and Levit, B.Y. (2003). Adaptive estimation of analytic functions on an interval., Math. Methods Statist. 12 62–94.
  • [3] Butucea, C. (2001). Exact adaptive pointwise estimation on Sobolev classes of densities., ESAIM Probab. Statist. 5 1–31.
  • [4] Butucea, C. (2004). Deconvolution of supersmooth densities with smooth noise., Canad. J. Statist. 32 181–192.
  • [5] Butucea, C. and Neumann, M.H. (2005). Exact asymptotics for estimating the marginal density of discretely observed diffusion processes., Bernoulli 11 411–444.
  • [6] Butucea, C. and Tsybakov, A.B. (2007). Sharp optimality for density deconvolution with dominating bias. I and II., Theory Probab. Appl. 52 24–39.
  • [7] Cai, T.T. and Low, M.G. (2005). Adaptive estimation of linear functionals under different performance measures., Bernoulli 11 341–358.
  • [8] Cai, T.T. and Low, M.G. (2005). On adaptive estimation of linear functionals., Ann. Statist. 33 2311–2343.
  • [9] Cator, E.A. (2001). Deconvolution with arbitrarily smooth kernels., Statist. Probab. Lett. 54 205–214.
  • [10] Cavalier, L. (2001). On the problem of local adaptive estimation in tomography., Bernoulli 7 63–78.
  • [11] Chaleyat-Maurel, M. and Genon-Catalot, V. (2006). Computable infinite-dimensional filters with applications to discretized diffusion processes., Stochastic Process. Appl. 116 1447–1467.
  • [12] Comte, F., Dedecker, J. and Taupin, M.-L. (2008). Adaptive density estimation for general ARCH models., Econometric Theory 24 1628–1662.
  • [13] Comte, F. and Genon-Catalot, V. (2006). Penalized projection estimator for volatility density., Scand. J. Statist. 33 875–893.
  • [14] Comte, F., Rozenholc, Y. and Taupin, M.-L. (2006). Penalized contrast estimator for adaptive density deconvolution., Canad. J. Statist. 34 431–452.
  • [15] Doukhan, P. (1994)., Mixing. Properties and examples. New York: Springer.
  • [16] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems., Ann. Statist. 19 1257–1272.
  • [17] Genon-Catalot, V. and Kessler, M. (2004). Random scale perturbation of an AR(1) process and its properties as a nonlinear explicit filter., Bernoulli 10 701–720.
  • [18] Goldenshluger, A. (1999). On pointwise adaptive nonparametric deconvolution., Bernoulli 5 907–925.
  • [19] Goldenshluger, A. and Pereverzev, S.V. (2003). On adaptive inverse estimation of linear functionals of Hilbert scales., Bernoulli 9 783–807.
  • [20] Golubev, G.K. (2004). The method of risk envelopes in the estimation of linear functionals., Problemy Peredachi Informatsii 40 58–72.
  • [21] Golubev, Y. and Levit, B. (2004). An oracle approach to adaptive estimation of linear functionals in a Gaussian model., Math. Methods Statist. 13 392–408.
  • [22] Klemelä, J. and Tsybakov, A.B. (2004). Exact constants for pointwise adaptive estimation under the Riesz transform., Probab. Theory Related Fields 129 441–467.
  • [23] Lacour, C. (2006). Rates of convergence for nonparametric deconvolution., C. R. Math. Acad. Sci. Paris 342 877–882.
  • [24] Laurent, B., Ludeña, C. and Prieur, C. (2008). Adaptive estimation of linear functionals by model selection., Electron. J. Stat. 2 993–1020.
  • [25] Lepski, O.V. and Levit, B.Y. (1998). Adaptive minimax estimation of infinitely differentiable functions., Math. Methods Statist. 7 123–156.
  • [26] Lepskiĭ, O.V. (1990). A problem of adaptive estimation in Gaussian white noise., Teor. Veroyatnost. i Primenen. 35 459–470.
  • [27] Petrov, V.V. (1995)., Limit Theorems of Probability Theory. New York: The Clarendon Press, Oxford Univ. Press.
  • [28] Tsybakov, A.B. (1998). Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes., Ann. Statist. 26 2420–2469.
  • [29] Van Es, B., Spreij, P. and van Zanten, H. (2005). Nonparametric volatility density estimation for discrete time models., J. Nonparametr. Statist. 17 237–251.