Bernoulli

  • Bernoulli
  • Volume 21, Number 4 (2015), 1984-2023.

Pointwise adaptive estimation of a multivariate density under independence hypothesis

Gilles Rebelles

Full-text: Open access

Abstract

In this paper, we study the problem of pointwise estimation of a multivariate density. We provide a data-driven selection rule from the family of kernel estimators and derive for it a pointwise oracle inequality. Using the latter bound, we show that the proposed estimator is minimax and minimax adaptive over the scale of anisotropic Nikolskii classes. It is important to emphasize that our estimation method adjusts automatically to eventual independence structure of the underlying density. This, in its turn, allows to reduce significantly the influence of the dimension on the accuracy of estimation (curse of dimensionality). The main technical tools used in our considerations are pointwise uniform bounds of empirical processes developed recently in Lepski [ Math. Methods Statist. 22 (2013) 83–99].

Article information

Source
Bernoulli, Volume 21, Number 4 (2015), 1984-2023.

Dates
Received: June 2013
Revised: March 2014
First available in Project Euclid: 5 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1438777584

Digital Object Identifier
doi:10.3150/14-BEJ633

Mathematical Reviews number (MathSciNet)
MR3378457

Zentralblatt MATH identifier
06502614

Keywords
adaptation density estimation independence structure oracle inequality upper function

Citation

Rebelles, Gilles. Pointwise adaptive estimation of a multivariate density under independence hypothesis. Bernoulli 21 (2015), no. 4, 1984--2023. doi:10.3150/14-BEJ633. https://projecteuclid.org/euclid.bj/1438777584


Export citation

References

  • [1] Birgé, L. (2008). Model selection for density estimation with ${\mathbb{L}}_{2}$-loss. Available at arXiv:0808.1416v2.
  • [2] Bretagnolle, J. and Huber, C. (1979). Estimation des densités: Risque minimax. Z. Wahrsch. Verw. Gebiete 47 119–137.
  • [3] Brown, L.D. and Low, M.G. (1996). A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. 24 2524–2535.
  • [4] Butucea, C. (2000). Two adaptive rates of convergence in pointwise density estimation. Math. Methods Statist. 9 39–64.
  • [5] Chacón, J.E. and Duong, T. (2010). Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices. TEST 19 375–398.
  • [6] Comte, F. and Lacour, C. (2013). Anisotropic adaptive kernel deconvolution. Ann. Inst. Henri Poincaré Probab. Stat. 49 569–609.
  • [7] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The $L_{1}$ View. Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics. New York: Wiley.
  • [8] Devroye, L. and Lugosi, G. (1996). A universally acceptable smoothing factor for kernel density estimates. Ann. Statist. 24 2499–2512.
  • [9] Devroye, L. and Lugosi, G. (1997). Nonasymptotic universal smoothing factors, kernel complexity and Yatracos classes. Ann. Statist. 25 2626–2637.
  • [10] Devroye, L. and Lugosi, G. (2001). Combinatorial Methods in Density Estimation. Springer Series in Statistics. New York: Springer.
  • [11] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508–539.
  • [12] Donoho, D.L. and Low, M.G. (1992). Renormalization exponents and optimal pointwise rates of convergence. Ann. Statist. 20 944–970.
  • [13] Efromovich, S. (2008). Adaptive estimation of and oracle inequalities for probability densities and characteristic functions. Ann. Statist. 36 1127–1155.
  • [14] Efromovich, S.Y. (1985). Non parametric estimation of a density of unknown smoothness. Theory Probab. Appl. 30 557–568.
  • [15] Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. Henri Poincaré Probab. Stat. 38 907–921.
  • [16] 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.
  • [17] Goldenshluger, A. and Lepski, O. (2013). On adaptive minimax density estimation on ${\mathbb{R}}^{D}$. Probab. Theory Related Fields. To appear. Published online 13 July 2013.
  • [18] Goldenshluger, A. and Lepski, O. (2011). Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. Ann. Statist. 39 1608–1632.
  • [19] Golubev, G.K. (1992). Nonparametric estimation of smooth densities of a distribution in $L_{2}$. Problemy Peredachi Informatsii 28 52–62.
  • [20] Hasminskii, R. and Ibragimov, I. (1990). On density estimation in the view of Kolmogorov’s ideas in approximation theory. Ann. Statist. 18 999–1010.
  • [21] Ibragimov, I.A. and Has’minskiĭ, R.Z. (1980). An estimate of the density of a distribution. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 98 61–85, 161–162, 166.
  • [22] Ibragimov, I.A. and Khas’minskiĭ, R.Z. (1981). More on estimation of the density of a distribution. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 108 72–88, 194, 198.
  • [23] Juditsky, A. and Lambert-Lacroix, S. (2004). On minimax density estimation on $\mathbb{R}$. Bernoulli 10 187–220.
  • [24] Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising. Probab. Theory Related Fields 121 137–170.
  • [25] Kerkyacharian, G., Lepski, O. and Picard, D. (2007). Nonlinear estimation in anisotropic multi-index denoising. Sparse case. Theory Probab. Appl. 52 150–171.
  • [26] Kerkyacharian, G., Picard, D. and Tribouley, K. (1996). $L^{p}$ adaptive density estimation. Bernoulli 2 229–247.
  • [27] Kluchnikoff, N. (2005). On adaptive estimation of anisotropic functions. Ph.D Thesis, Aix-Marseille 1.
  • [28] Lepski, O. (2013). Upper functions for positive random functionals. II. Application to the empirical processes theory, part 1. Math. Methods Statist. 22 83–99.
  • [29] Lepski, O. (2013). Multivariate density estimation under sup-norm loss: Oracle approach, adaptation and independence structure. Ann. Statist. 41 1005–1034.
  • [30] Lepski, O.V., Mammen, E. and Spokoiny, V.G. (1997). Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 929–947.
  • [31] Lepskiĭ, O.V. (1991). A problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454–466.
  • [32] Mason, D.M. (2009). Risk bounds for kernel density estimators. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 363 66–104, 183.
  • [33] Massart, P. (2007). Concentration Inequalities and Model Selection. Lecture Notes in Math. 1896. Berlin: Springer.
  • [34] Nikol’skiĭ, S.M. (1977). Priblizhenie Funktsii Mnogikh Peremennykh i Teoremy Vlozheniya, 2nd ed. Moscow: Nauka.
  • [35] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065–1076.
  • [36] Rigollet, P. (2006). Adaptive density estimation using the blockwise Stein method. Bernoulli 12 351–370.
  • [37] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 832–837.
  • [38] 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. Hackensack, NJ: World Sci. Publ.
  • [39] Scott, D.W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
  • [40] Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. Monographs on Statistics and Applied Probability. London: Chapman & Hall.
  • [41] Tsybakov, A.B. (1998). Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. Ann. Statist. 26 2420–2469.