Sankhya A

, Volume 76, Issue 1, pp 48–76 | Cite as

On the estimation of density-weighted average derivative by wavelet methods under various dependence structures

  • Christophe ChesneauEmail author
  • Maher Kachour
  • Fabien Navarro


The problem of estimating the density-weighted average derivative of a regression function is considered. We present a new consistent estimator based on a plug-in approach and wavelet projections. Its performances are explored under various dependence structures on the observations: the independent case, the ρ-mixing case and the α-mixing case. More precisely, denoting n the number of observations, in the independent case, we prove that it attains 1/n under the mean squared error, in the ρ-mixing case, \(1/\sqrt{n}\) under the mean absolute error, and, in the α-mixing case, \(\sqrt{\ln n /n}\) under the mean absolute error. A short simulation study illustrates the theory.

Keywords and phrases.

Nonparametric estimation of density-weighted average derivative ‘plug-in’ approach, wavelets  consistency ρ-mixing α-mixing 

AMS (2000) subject classification.

Primary 62G08 Secondary 62G20 Secondary 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Antoniadis, A. (1997). Wavelets in statistics: a review (with discussion). J. Ital. Statist. Soc., 6, 97–144.CrossRefGoogle Scholar
  2. Banerjee, A.N. (2007). A method of estimating the average derivative. J. Econometrics, 136, 65–88.CrossRefMathSciNetGoogle Scholar
  3. Cattaneo, M.D., Crump, R.K. and Jansson, M. (2010). Robust data-driven inference for density-weighted average derivatives. J. Amer. Stat. Assoc., 105, 1070– 1083.CrossRefMathSciNetGoogle Scholar
  4. Cattaneo, M.D., Crump, R.K. and Jansson, M. (2008). Small bandwidth asymptotics for density-weighted average derivatives. CREATES Research Paper 200824. Available at SSRN:
  5. Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory, 18, 17–39.CrossRefzbMATHMathSciNetGoogle Scholar
  6. Chaubey, Y.P. and Doosti, H. (2005). Wavelet based estimation of the derivatives of a density for m-dependent random variables. J. Iran. Stat. Soc., 4, 97–105.Google Scholar
  7. Chaubey, Y.P., Doosti, H. and Prakasa Rao, B.L.S. (2006). Wavelet based estimation of the derivatives of a density with associated variables. Int. J. Pure Appl. Math., 27, 97–106.zbMATHMathSciNetGoogle Scholar
  8. Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1993). Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal., 24, 54–81.CrossRefGoogle Scholar
  9. Coppejans, M. and Sieg, H. (2005). Kernel Estimation of Average Derivatives and Differences. J. Bus. Econ. Stat., 23, 211–225.CrossRefMathSciNetGoogle Scholar
  10. Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF regional conferences series in applied mathematics. SIAM, Philadelphia.Google Scholar
  11. Davydov, Y. (1970). The invariance principle for stationary processes. Theor. Probab. Appl., 15, 498–509.CrossRefzbMATHGoogle Scholar
  12. Deaton, A. and Ng, S. (1998). Parametric and nonparametric approaches to price and tax reform. J. Amer. Stat. Assoc., 93, 900–909.CrossRefGoogle Scholar
  13. Doukhan, P. (1994). Mixing. Properties and Examples. Lecture Notes in Statistics 85. Springer Verlag, New York.zbMATHGoogle Scholar
  14. Fryzlewicz, P. and Subba Rao, S. (2011). Mixing properties of ARCH and time-varying ARCH processes. Bernoulli, 17, 320–346.CrossRefzbMATHMathSciNetGoogle Scholar
  15. Härdle, W. and Stoker, T.M. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Stat. Assoc., 84, 986–995.zbMATHGoogle Scholar
  16. Härdle, W., Hildenbrand, W. and Jerison, M. (1991). Empirical evidence on the law of demand. Econometrica, 59, 1525–1549.CrossRefzbMATHGoogle Scholar
  17. Härdle, W., Hart, J., Marron, J.S. and Tsybakov, A.B. (1992). Bandwidth choice for average derivative estimation. J. Amer. Stat. Assoc., 87, 417, 218–226.zbMATHGoogle Scholar
  18. Härdle, W. and Tsybakov, A.B. (1993). How sensitive are average derivatives?. J. Econom., 58, 31–48.CrossRefzbMATHGoogle Scholar
  19. Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelet, Approximation and Statistical Applications. Lectures Notes in Statistics 129. Springer Verlag, New York.CrossRefGoogle Scholar
  20. Hansen, B. (2009). Lecture Notes on Nonparametrics, Lecture Notes.Google Scholar
  21. Juditsky, A. and Lambert-Lacroix, S. (2004). On minimax density estimation on R. Bernoulli, 10, 187–220.CrossRefzbMATHMathSciNetGoogle Scholar
  22. Kolmogorov, A.N. and Rozanov, Yu.A. (1960). On strong mixing conditions for stationary Gaussian processes. Theor. Probab. Appl., 5, 204–208.CrossRefMathSciNetGoogle Scholar
  23. Lütkepohl, H. (1992). Multiple Time Series Analysis. Springer-Verlag, Heidelberg.Google Scholar
  24. Mallat, S. (2009). A Wavelet Tour of Signal Processing, Third Edition. The Sparse Way, with Contributions from Gabriel Peyré. Elsevier/Academic Press, Amsterdam.Google Scholar
  25. Marron, J.S., Adak, S., Johnstone, I.M., Neumann, M.H. and Patil, P. (1998). Exact risk analysis of wavelet regression. J. Comput. Graph. Statist., 7, 278–309.Google Scholar
  26. Matzkin, R.L. (2007). Nonparametric Identification. In Handbook of Econometrics (J. Heckman and E. Leamer Eds.). Elsevier Science B.V., VIB, pp. 5307– 5368.Google Scholar
  27. Meyer, Y. (1992). Wavelets and Operators. Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  28. Powell, J.L., Stock, J.H. and Stoker, T.M. (1989). Semiparametric estimation of index coefficients. Econometrica, 57, 1403–1430.CrossRefzbMATHMathSciNetGoogle Scholar
  29. Powell, J.L. (1994). Estimation of Semiparametric Models. In Handbook of Econometrics (R. Engle and D. McFadden Eds.). Elsevier Science B.V., IV, pp. 2443– 2521.Google Scholar
  30. Powell, J.L. and Stoker, T.M. (1996). Optimal bandwidth choice for density-weighted averages. J. Econometrics, 75, 291–316.CrossRefzbMATHMathSciNetGoogle Scholar
  31. Prakasa Rao, B.L.S. (1995). Consistent estimation of density-weighted average derivative by orthogonal series method. Statist. Probab. Lett., 22, 205–212.CrossRefzbMATHMathSciNetGoogle Scholar
  32. Prakasa Rao, B.L.S. (1996). Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cybernet., 28, 91–10.MathSciNetGoogle Scholar
  33. Reynaud-Bouret, P., Rivoirard, V. and Tuleau-Malot, C. (2011). Adaptive density estimation: a curse of support? J. Statist. Plann. Inference, 141, 115–139.CrossRefzbMATHMathSciNetGoogle Scholar
  34. Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA, 42, 43–47.CrossRefzbMATHMathSciNetGoogle Scholar
  35. Rosenthal, H.P. (1970). On the subspaces of \(\mathbb{L}_p\) (p ≥ 2) spanned by sequences of independent random variables. Israel J. Math., 8, 273–303.CrossRefzbMATHMathSciNetGoogle Scholar
  36. Schafgans, M. and Zinde-Walsh, V. (2010). Smoothness adaptive average derivative estimation. Econom. J., 13, 40–62.CrossRefzbMATHMathSciNetGoogle Scholar
  37. Shao, Q.-M. (1995). Maximal inequality for partial sums of ρ-mixing sequences. Ann. Probab., 23, 948–965.CrossRefzbMATHMathSciNetGoogle Scholar
  38. Stoker, T.M. (1986). Consistent estimation of scaled coefficients. Econometrica, 54, 1461–1481.CrossRefzbMATHMathSciNetGoogle Scholar
  39. Stoker, T.M. (1989). Tests of additive derivative constraints. Rev. Econ. Stud., 56, 535–552.CrossRefzbMATHMathSciNetGoogle Scholar
  40. Stoker, T.M. (1991). Equivalence of Direct, Indirect and Slope Estimators of Average Derivatives, Nonparametric and Semiparametric Methods in Econometrics and Statistics (W.A. Barnett, J. Powell and G. Tauchen Eds.). Cambridge University Press.Google Scholar
  41. Türlach, B.A. (1994). Fast implementation of density-weighted average derivative estimation. Computationally Intensive Statistical Methods, 26, 28–33.Google Scholar
  42. White, H. and Domowitz, I. (1984). Nonlinear regression with dependent observations. Econometrica, 52, 143–162.CrossRefzbMATHMathSciNetGoogle Scholar
  43. Vidakovic, B. (1999). Statistical Modeling by Wavelets. John Wiley & Sons, Inc., New York, 384 pp.CrossRefzbMATHGoogle Scholar
  44. Zhengyan, L. and Lu, C. (1996). Limit Theory for Mixing Dependent Random Variables. Kluwer, Dordrecht.zbMATHGoogle Scholar

Copyright information

© Indian Statistical Institute 2013

Authors and Affiliations

  • Christophe Chesneau
    • 1
    Email author
  • Maher Kachour
    • 2
  • Fabien Navarro
    • 1
    • 3
  1. 1.Département de Mathématiques, UFR de Sciences, LMNOUniversité de Caen Basse-NormandieCaen CedexFrance
  2. 2.École supérieure de commerce IDRACLyon Cedex 09France
  3. 3.GREYC CNRS-ENSICAEN-Université de CaenCaen CedexFrance

Personalised recommendations