Electronic Journal of Statistics

$\mathbb{L}_{p}$ adaptive estimation of an anisotropic density under independence hypothesis

Gilles Rebelles

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In this paper, we focus on the problem of a multivariate density estimation under an $\mathbb{L}_{p}$-loss. We provide a data-driven selection rule from a family of kernel estimators and derive for it $\mathbb{L}_{p}$-risk oracle inequalities depending on the value of $p\geq1$. The proposed estimator permits us to take into account approximation properties of the underlying density and its independence structure simultaneously. Specifically, we obtain adaptive upper bounds over a scale of anisotropic Nikolskii classes when the smoothness is also measured with the $\mathbb{L}_{p}$-norm. It is important to emphasize that the adaptation to unknown independence structure of the estimated density allows us to improve significantly the accuracy of estimation (curse of dimensionality). The main technical tools used in our derivation are uniform bounds on the $\mathbb{L}_{p}$-norms of empirical processes developed in Goldenshluger and Lepski [13].

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Electron. J. Statist., Volume 9, Number 1 (2015), 106-134.

First available in Project Euclid: 6 February 2015

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Density estimation oracle inequality adaptation independence structure upper function


Rebelles, Gilles. $\mathbb{L}_{p}$ adaptive estimation of an anisotropic density under independence hypothesis. Electron. J. Statist. 9 (2015), no. 1, 106--134. doi:10.1214/15-EJS986. https://projecteuclid.org/euclid.ejs/1423229752

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  • [1] Baraud, Y. and Birgé, L. (2014). Estimating composite functions by model selection., Ann. Inst. H. Poincaré Probab. Statist. 50, 285–314.
  • [2] Birgé, L. (2013). Model selection for density estimation with $\mathbbL_2$-loss., Probab. Theory and Relat. Fields 158, 533–574.
  • [3] Bretagnolle, J. and Huber, C. (1979). Estimation des densités: risque minimax., Z. Wahrsch. Verw. Gebiete 47, 119–137.
  • [4] Chacón, J.E. and Duong, T. (2010). Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices., Test 19, 375–398.
  • [5] Donoho, D.L., Johnston, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding., Ann. Statist. 24, 508–539.
  • [6] Devroye, L. and Györfi, L. (1985)., Nonparametric Density Estimation: The $\mathbbL_1$ View. Wiley Series in Probability and Statistics. Wiley, New York.
  • [7] Devroye, L. and Lugosi, G. (1996). A universally acceptable smoothing factor for kernel density estimation., Ann. Statist. 24, 2499–2512.
  • [8] Devroye, L. and Lugosi, G. (1997). Nonasymptotic universal smoothing factor, kernel complexity and Yatracos classes., Ann. Statist. 25, 2626–2637.
  • [9] Devroye, L. and Lugosi, G. (2001)., Combinatorial Methods in Density Estimation. Springer, New York.
  • [10] Efromovich, S.Y. (1985). Nonparametric estimation of a density of unknown smoothness., Theory. Probab. Appl. 30, 557–568.
  • [11] Efromovich, S.Y. (2008). Adaptive estimation of and oracle inequalities for probability densities and characteristic functions., Ann. Statist. 36, 1127–1155.
  • [12] Goldenshluger, A. and Lepski, O.V. (2009). Structural adaptation via $\mathbbL_p$-norm oracle inequalities., Probab. Theory Related Fields, 143, 41–71.
  • [13] Goldenshluger, A. and Lepski, O. (2011a). Uniform bounds for norms of sums of independent random functions., Ann. Probab. 39, 2318–2384.
  • [14] Goldenshluger, A. and Lepski, O. (2011b). Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality., Ann. Statist. 39, 1608–1632.
  • [15] Goldenshluger, A. and Lepski, O. (2013). On adaptive minimax density estimation on $\mathbbR^d$., Probab. Theory and Relat. Fields 159, 479–543.
  • [16] Golubev, G.K. (1992). Nonparametric estimation of smooth probability densities., Probl. Inf. Transm. 1, 52–62.
  • [17] Hasminskii, R. and Ibragimov, I. (1990). On density estimation in the view of Kolmogorov’s ideas in approximation theory., Ann. Statist. 18, 999–1010.
  • [18] Horowitz, J.I. and Mamen, I. (1990). Rate-optimal estimation for a general class of nonparametric regression models with unknown link functions., Ann. Statist. 35, 6, 2589–2619.
  • [19] Ibragimov, I.A. and Khasminski, R.Z. (1980). An estimate of the density of a distribution., Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 98, 61–85 (in Russian).
  • [20] Ibragimov, I.A. and Khasminski, R.Z. (1981). More on estimation of the density of a distribution., Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 108, 72–88 (in Russian).
  • [21] Iouditski, A.B., Lepski, O.V. and Tsybakov, A.B. (2009). Nonparametric estimation of composite functions., Ann. Statist. 37, 3, 1360–1440.
  • [22] Juditsky, A. and Lambert-Lacroix, S. (2004). On minimax density estimation on $\mathbbR$., Bernoulli 10, 187–220.
  • [23] Kerkyacharian, G., Picard, D. and Tribouley, K. (1996). $L^p$ adaptive density estimation., Bernoulli 2, 229–247.
  • [24] Kerkyacharian, G., Lepski, O.V. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising., Probab. Theory and Relat. Fields 121, 137–170.
  • [25] Kerkyacharian, G., Lepski, O.V. and Picard, D. (2007). Nonlinear estimation in anisotropic multi-index denoising. Sparse case., Probab. Theory Appl. 52, 150–171.
  • [26] Lepski, O.V. (1991). On a problem of adaptive estimation in Gaussian white noise model., Theory Probab. Appl. 35, 454–466.
  • [27] Lepski, O.V. (2013). Multivariate density estimation under sup-norm loss: oracle approach, adaptation and independence structure., Ann. Stat. 40, 2, 1005–1034.
  • [28] Lepski, O.V. and Serdyukova, N. (2014). Adaptive estimation under single-index constraint in a regression model., Ann. Stat. 40, 1, 1–28.
  • [29] Mason, D.M. (2009). Risk bounds for density kernel estimators., Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 363, 66–104. Available at http://www.pdmi.ras.ru/znsl/.
  • [30] Massart, P. (2007)., Concentration Inequalities and Model Selection. Lecture from the 33rd Summer School on Probability Theory held in Saint-Floor, July 6–23, 2003. Lecture Notes in Mathematics, 1886. Springer, Berlin.
  • [31] Nikol’skii, S.M. (1977). Priblizheni Funktsii Mnogikh Peremennykh i Teoremy Vlosheniya (in Russian). [Approximation of functions of several variables and imbedding theorems.] Naukka, Moscow, 1977.
  • [32] Parzen, E. (1962). On the estimation of a probability density function and the mode., Ann. Math. Statist. 33, 1065–1076.
  • [33] Rebelles, G. (2014). Pointwise adaptive estimation of a multivariate density under independence hypothesis., Bernoulli, forthcoming.
  • [34] Rigollet, P.H. (2006). Adaptive density estimation using the blockwise Stein method., Bernoulli 12, 351–370.
  • [35] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function., Ann. Math. Statist. 27, 832–837.
  • [36] Samarov, A. and Tsybakov, A.B. (2007). Aggregation of density estimators and dimension reduction., Advances in Statistical Modeling and inference, 233–251, Ser. Biostat., 3, World Sci. Publ., Hackensack, NJ.
  • [37] Silverman, B.W. (1986)., Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
  • [38] Scott, D. (1992)., Multivariate Density Estimation. Wiley, New York.
  • [39] Tsybakov, A.B. (2009)., Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, New York.