Sobolev-Hermite versus Sobolev nonparametric density estimation on \({\mathbb {R}}\)

  • Denis Belomestny
  • Fabienne ComteEmail author
  • Valentine Genon-Catalot


In this paper, our aim is to revisit the nonparametric estimation of a square integrable density f on \({\mathbb {R}}\), by using projection estimators on a Hermite basis. These estimators are studied from the point of view of their mean integrated squared error on \({\mathbb {R}}\). A model selection method is described and proved to perform an automatic bias variance compromise. Then, we present another collection of estimators, of deconvolution type, for which we define another model selection strategy. Although the minimax asymptotic rates of these two types of estimators are mainly equivalent, the complexity of the Hermite estimators is usually much lower than the complexity of their deconvolution (or kernel) counterparts. These results are illustrated through a small simulation study.


Complexity Density estimation Hermite basis Model selection Projection estimator 


  1. Abramowitz, M., Stegun, I. A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55 for sale by the Superintendent of Documents. Washington, DC: U.S. Government Printing Office.Google Scholar
  2. Barron, A., Birgé, L., Massart, P. (1999). Risk bounds for model selection via penalization. Probability Theory and Related Fields, 113, 301–413.Google Scholar
  3. Belomestny, D., Comte, F., Genon-Catalot, V., Masuda, H., Reiss, M. (2015). Lévy matters. IV. Estimation for discretely observed Lévy processes. Lecture Notes in Mathematics, 2128. Lévy Matters. Cham: Springer.Google Scholar
  4. Belomestny, D., Comte, F., Genon-Catalot, V. (2016). Nonparametric Laguerre estimation in the multiplicative censoring model. Electronic Journal of Statistics, 10, 3114–3152.Google Scholar
  5. Bertoin, J. (1996). Lévy processes. Cambridge tracts in mathematics, 121. Cambridge, NY: Cambridge University Press.Google Scholar
  6. Birgé, L., Massart, P. (2007). Minimal penalties for Gaussian model selection. Probability Theory and Related Fields, 138, 33–73.Google Scholar
  7. Bongioanni, B., Torrea, J. L. (2006). Sobolev spaces associated to the harmonic oscillator. Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 116(3), 337–360.Google Scholar
  8. Chaleyat-Maurel, M., Genon-Catalot, V. (2006). Computable infinite-dimensional filters with applications to discretized diffusion processes. Stochastic Processes and Their Applications, 116, 1447–1467.Google Scholar
  9. Comte, F., Genon-Catalot, V. (2009). Nonparametric estimation for pure jump Lévy processes based on high frequency data. Stochastic Processes and Their Applications, 119, 4088–4123.Google Scholar
  10. Comte, F., Genon-Catalot, V. (2015). Adaptive Laguerre density estimation for mixed Poisson models. Electronic Journal of Statistics, 9, 1113–1149.Google Scholar
  11. Comte, F., Dedecker, J., Taupin, M. L. (2008). Adaptive density deconvolution with dependent inputs. Mathematical Methods of Statistics, 17, 87–112.Google Scholar
  12. Devroye, L., Györfi, L. (1985). Nonparametric density estimation. The L1 view. Wiley series in probability and mathematical statistics: Tracts on probability and statistics. New York: Wiley.Google Scholar
  13. Devroye, L., Lugosi, G. (2001). Combinatorial methods in density estimation. Springer series in statistics. New York: Springer.Google Scholar
  14. Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., Picard, D. (1996). Density estimation by wavelet thresholding. The Annals of Statistics, 24, 508–539.Google Scholar
  15. Efromovich, S. (1999). Nonparametric curve estimation. Methods, theory, and applications. Springer series in statistics. New York: Springer.Google Scholar
  16. Efromovich, S. (2008). Adaptive estimation of and oracle inequalities for probability densities and characteristic functions. The Annals of Statististics, 36, 1127–1155.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Efromovich, S. (2009). Lower bound for estimation of Sobolev densities of order less 1/2. Journal of Statistical Planning and Inference, 139, 2261–2268.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Ibragimov, I. (2001). Estimation of analytic functions. In M. de Gunst, C. Klaasen, A. van der Vaart (Eds.) State of the art in probability and statistics (Leiden, 1999). Institute of mathematical statistics lecture notes—Monograph series 36 (pp. 359–383). Beachwood, OH: Institute of Mathematical Statistics.Google Scholar
  19. Ibragimov, I. A., Has’minskii, R. Z. (1980). An estimate of the density of a distribution. Studies in mathematical statistics, IV. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI) 98, 61–85, 161–162 (in Russian).Google Scholar
  20. Kim, A. K. H. (2014). Minimax bounds for estimation of normal mixtures. Bernoulli, 20, 1802–1818.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Klein, T., Rio, E. (2005). Concentration around the mean for maxima of empirical processes. The Annals of Probability, 33, 1060–1077.Google Scholar
  22. Lebedev, N. N. (1972). Special functions and their applications (Revised edition, translated from the Russian and edited by Richard A. Silverman. Unabridged and corrected republication). New York: Dover Publications, Inc.Google Scholar
  23. Mabon, G. (2017). Adaptive deconvolution on the nonnegative real line. Scandinavian Journal of Statistics: Theory and Applications, 44, 707–740.Google Scholar
  24. Markett, C. (1984). Norm estimates for \((C,\delta )\) means of Hermite expansions and bounds for \(\delta _{{\rm eff}}\). Acta Mathematica Hungarica, 43, 187–198.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Massart, P. (2007). Concentration inequalities and model selection. Lectures from the 33rd summer school on probability theory held in Saint-Flour, July 6–23, 2003. With a foreword by Jean Picard. Lecture notes in mathematics, 1896. Berlin: Springer.Google Scholar
  26. Rigollet, P. (2006). Adaptive density estimation using the blockwise Stein method. Bernoulli, 12, 351–370.MathSciNetCrossRefzbMATHGoogle Scholar
  27. Schipper, M. (1996). Optimal rates and constants in L2-minimax estimation of probability density functions. Mathematical Methods of Statistics, 5, 253–274.MathSciNetzbMATHGoogle Scholar
  28. Schwartz, S. C. (1967). Estimation of a probability density by an orthogonal series. The Annals of Mathematical Statistics, 38, 1261–1265.MathSciNetCrossRefzbMATHGoogle Scholar
  29. Szegö, G. (1975). Orthogonal polynomials (4th ed.). American Mathematical Society, Colloquium Publications, Vol: XXIII. Providence, RI: American mathematical Society.Google Scholar
  30. Tsybakov, A. B. (2009). Introduction to nonparametric estimation. Springer series in statistics. New York: Springer.Google Scholar
  31. Walter, G. G. (1977). Properties of Hermite series estimation of probability density. The Annals of Statistics, 5, 1258–1264.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  • Denis Belomestny
    • 1
    • 2
  • Fabienne Comte
    • 3
    Email author
  • Valentine Genon-Catalot
    • 3
  1. 1.Faculty of MathematicsDuisburg-Essen UniversityEssenGermany
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.MAP5 UMR CNRS 8145Université Paris Descartes, Sorbonne Paris CitéParis Cedex 06France

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