Skip to main content
SpringerLink
Log in

L p-consistency of multivariate density estimates

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

L p notion of the weak, mean, and strong consistency of the kernel method of multivariate density estimation is proposed and studied. The results expand, unify, or generalize most known results in the literature. Rates of convergence in mean and strongL p-consistencies are presented.

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. Cacoullos, T. (1966). Estimation of a multivariate density,Ann. Inst. Statist. Math.,18, 179–189.

    Article  MathSciNet  Google Scholar 

  2. Dharmadhikari, S. W., Fabian, V. and Jogdeo, K. (1968). Bounds on the moments of martingales,Ann. Math. Statist.,39, 1719–1723.

    Article  MathSciNet  Google Scholar 

  3. Fryer, M. J. (1977). A review of some nonparametric methods of density estimation.J. Inst. Math. Appl.,20, 335–354.

    Article  MathSciNet  Google Scholar 

  4. Kuelbs, J. (1978). Some exponential moments of sums of independent random variables,Trans. Amer. Math. Soc.,240, 145–162.

    Article  MathSciNet  Google Scholar 

  5. Leadbetter, M. R. (1963). On the nonparametric estimation of probability densities,Tech. Report No. 11, Research Triangle Institute.

  6. Loeve, M. (1963).Probability Theory, Von Nostrand.

  7. Matusita, K. (1955). Decision rules based on the distance for the problem of fit, two samples, and estimation,Ann. Math. Statist.,26, 631–640.

    Article  MathSciNet  Google Scholar 

  8. Nadaraya, E. A. (1965). On nonparametric estimates of density functions and regression curves,Theory Prob. Appl.,10, 186–190.

    Article  Google Scholar 

  9. Parzen, E. (1962). On estimation of a probability density and mode,Ann. Math. Statist.,33, 1065–1076.

    Article  MathSciNet  Google Scholar 

  10. Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function,Ann. Math. Statist.,27, 832–837.

    Article  MathSciNet  Google Scholar 

  11. Rüschendorf, L. (1977). Consistency of estimators for multivarite density functions and for the mode,Sankhyã, A39, 243–250.

    MATH  Google Scholar 

  12. Schuster, E. F. (1970). Note on the uniform convergence of density estimates,Ann. Math. Statist.,41, 1347–1348.

    Article  MathSciNet  Google Scholar 

  13. Van Ryzin, J. (1969). On strong consistency of density estimates,Ann. Math. Statist.,41, 1765–1772.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Petroleum and Minerals, Saudi Arabia

    Ibrahim A. Ahmad

Authors
  1. Ibrahim A. Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Ahmad, I.A. L p-consistency of multivariate density estimates. Ann Inst Stat Math 34, 457–466 (1982). https://doi.org/10.1007/BF02481044

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02481044

AMS 1970 subject classification

Key words and phrases

Search

Navigation