Skip to main content
SpringerLink
Log in

Estimating a density under pointwise constraints on the derivatives

  • Published:
Mathematical Methods of Statistics Aims and scope Submit manuscript

Abstract

Suppose we want to estimate a density at a point where we know the values of its first or higher order derivatives. In this case a given kernel estimator of the density can be modified by adding appropriately weighted kernel estimators of these derivatives. We give conditions under which the modified estimators are asymptotically normal. We also determine the optimal weights. When the highest derivative is known to vanish at a point, then the bias is asymptotically negligible at that point and the asymptotic variance of the kernel estimator can be made arbitrarily small by choosing a large bandwidth.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. K. Bhattacharya, “Estimation of a Probability Density Function and Its Derivatives”, Sankhyā Ser. A 29, 373–382 (1967).

    MATH  Google Scholar 

  2. R. C. Bradley, “Asymptotic Normality of Some Kernel-Type Estimators of Probability Density”, Statist. Probab. Lett. 1, 295–300 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Fan and Q. Yao, Nonlinear Time Series. Nonparametric and Parametric Methods, in Springer Series in Statist. (Springer-Verlag, New York, 2003).

    Google Scholar 

  4. B. Y. Levit, “Conditional Estimation of Linear Functionals”, Probl. Inf. Transm. 11, 291–301 (1975).

    MathSciNet  Google Scholar 

  5. Z. Lu, “Asymptotic Normality of Kernel Density Estimators under Dependence”, Ann. Inst. Statist. Math. 53, 447–468 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Masry and J. Fan, “Local Polynomial Estimation of Regression Functions for Mixing Processes”, Scand. J. Statist. 24, 165–179 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  7. U. U. Müller and W. Wefelmeyer, “Estimators for Models with Constraints Involving Unknown Parameters”, Math. Methods Statist. 11, 221–235 (2002).

    MathSciNet  MATH  Google Scholar 

  8. E. Parzen, “On Estimation of a Probability Density Function and Mode”, Ann. Math. Statist. 33, 1065–1076 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Rosenblatt, “Remarks on Some Nonparametric Estimates of a Density Function”, Ann. Math. Statist. 27, 832–837 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  10. E. F. Schuster, “Estimation of a Probability Density Function and Its Derivatives”, Ann. Math. Statist. 40, 1187–1195 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  11. R. S. Singh, “Improvement on Some Known Nonparametric Uniformly Consistent Estimators of Derivatives of a Density”, Ann. Statist. 5, 394–399 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  12. R. S. Singh, “On the Exact Asymptotic Behavior of Estimators of a Density and Its Derivatives”, Ann. Statist. 9, 453–456 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. W. van der Vaart, Asymptotic Statistics, in Cambridge Series in Statist. and Probab. Math. 3 (Cambridge Univ. Press, Cambridge, 1998).

    Google Scholar 

  14. B. Zhang, “A Note on Kernel Density Estimation with Auxiliary Information”, Comm. Statist. Theory Methods 27, 1–11 (1998).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Texas A&M University, College Station, USA

    U. Müller

  2. Universität zu Köln, Köln, Germany

    W. Wefelmeyer

Authors
  1. U. Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Wefelmeyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to U. Müller.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Müller, U., Wefelmeyer, W. Estimating a density under pointwise constraints on the derivatives. Math. Meth. Stat. 23, 201–209 (2014). https://doi.org/10.3103/S106653071403003X

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S106653071403003X

Keywords

2000 Mathematics Subject Classification

Search

Navigation