The Annals of Statistics

Estimating nonquadratic functionals of a density using Haar wavelets

Gérard Kerkyacharian and Dominique Picard

Full-text: Open access

Abstract

Z.Consider the problem of estimating $\int \Phi(f)$, where $\Phi$ is a smooth function and f is a density with given order of regularity s. Special attention is paid to the case $\Phi(t) = t^3$. It has been shown that for low values of s the $n^{-1/2}$ rate of convergence is not achievable uniformly over the class of objects of regularity s. In fact, a lower bound for this rate is $n^{-4s/(1+4s)}$ for $0 < s \leq 1/4$. As for the upper bound, using a Taylor expansion, it can be seen that it is enough to provide an estimate for the case $\Phi(x) = x^3$. That is the aim of this paper. Our method makes intensive use of special algebraic and wavelet properties of the Haar basis.

Article information

Source
Ann. Statist., Volume 24, Number 2 (1996), 485-507.

Dates
First available in Project Euclid: 24 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032894450

Digital Object Identifier
doi:10.1214/aos/1032894450

Mathematical Reviews number (MathSciNet)
MR1394973

Zentralblatt MATH identifier
0860.62033

Subjects
Primary: G2G05 G2G20

Keywords
Minimax estimation estimation of nonlinear functionals integral functionals of a density wavelet estimate $U$-statistic

Citation

Kerkyacharian, Gérard; Picard, Dominique. Estimating nonquadratic functionals of a density using Haar wavelets. Ann. Statist. 24 (1996), no. 2, 485--507. doi:10.1214/aos/1032894450. https://projecteuclid.org/euclid.aos/1032894450


Export citation

References

  • BERGH, J. and LOFSTROM, J. 1976. Interpolation Spaces: An Introduction. Springer, Berlin. ¨ ¨ Z.
  • BICKEL, P. and RITOV, Y. 1988. Estimating integrated squares density derivatives: sharp best order of convergence estimates. Sankhy a Ser. A 50 381 393. Z.
  • BIRGE, L. and MASSART, P. 1995. Estimation of integral functionals of a density. Ann. Statist. ´ 23 11 29. Z.
  • DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. 1995. Wavelet shrinkage: Z. asy mptopia? with discussion. J. Roy. Statist. Soc. Ser. B 57 301 369. Z.
  • DONOHO, D. and NUSSBAUM, M. 1990. Minimax quadratic estimation of a quadratic functional. J. Complexity 6 290 323.
  • GOLDSTEIN, L. and MESSER, K. 1992. Optimal plug-in estimate for nonparametric functional estimation. Ann. Statist. 20 1306 1328. Z.
  • HALL, P. and MARRON, S. 1987. Estimation of integrated squared density derivatives. Statist. Probab. Lett. 6 109 115.Z.
  • HASMINSKII, R. and IBRAGIMOV, I. 1979. On the non parametric estimation of functionals. In Z Proceedings of the Second Prague Sy mposium in Asy mptotic Statistics P. Mandl and. M. Hushkova, eds. 41 55. North-Holland, Amsterdam. Z.
  • JOHNSTONE, I., KERKy ACHARIAN, G. and PICARD, D. 1992. Estimation d'une densite de proba´ bilite par methode d'ondelettes. C. R. Acad. Sci. Paris Ser. I Math. 315 211 216. ´ ´ ´ Z.
  • KERKy ACHARIAN, G. and PICARD, D. 1992a. Density estimation in Besov spaces. Statist. Probab. Lett. 13 15 24. Z.
  • KERKy ACHARIAN, G. and PICARD, D. 1992b. Density estimation by kernel and wavelet method: link between kernel geometry and regularity constraints. C. R. Acad. Sci. Paris Ser. I ´ Math. 315 79 84. Z.
  • KERKy ACHARIAN, G. and PICARD, D. 1993. Density estimation by kernel and wavelet methods: optimality of Besov spaces. Statist. Probab. Lett. 18 327 336. Z.
  • LAURENT, B. 1996. Efficient estimation of integral functionals of a density. Ann. Statist. 24 659 681. Z.
  • LEVIT, B. 1979. Asy mptotically efficient estimation of non linear functionals. Problems Inform. Transmission 14 65 72. Z.
  • MEy ER, Y. 1990. Ondelettes et Operateurs I. Hermann, Paris. ´ Z.
  • NEMIROVSKII, A. S. 1990. On necessary conditions for efficient estimation of functional of a non parametric signal in white noise model. Theory Probab. Appl. 35 94 103. Z.
  • PEETRE, J. 1975. New Thoughts on Besov Spaces. Dept. Mathematics, Duke Univ., Durham, NC. Z.
  • RITOV, Y. and BICKEL, P. 1990. Achieving information bounds in non and semi parametric models. Ann. Statist. 18 925 938.