Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Properties of nonparametric estimators of autocovariance for stationary random fields
Download PDF
Download PDF
  • Published: September 1994

Properties of nonparametric estimators of autocovariance for stationary random fields

  • Peter Hall1 nAff2 &
  • Prakash Patil1 

Probability Theory and Related Fields volume 99, pages 399–424 (1994)Cite this article

  • 484 Accesses

  • 81 Citations

  • 3 Altmetric

  • Metrics details

Summary

We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovatiance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • Armstrong, A.G., Diamond, P.: Testing variograms for positive-definiteness. Math. Geol.16, 407–421 (1984)

    Google Scholar 

  • Cheng, K.F., Lin, P.E.: Nonparametric estimation of a regression function. Z. Wahrscheinlichkeitstheor. Verw. Geb.57, 223–233 (1981)

    Google Scholar 

  • Christakos, G.: On the problem of permissible covariance and variogram models. Water Resources Res.20, 251–265 (1984)

    Google Scholar 

  • Cressic, N.: Statistics for Spatial Data. New York: Wiley, 1991

    Google Scholar 

  • Devroye, L., Györfi, L.: Nonparametric Density Estimation: TheL 1 View. New York: Wiley, 1985

    Google Scholar 

  • Gasser, Th., Müller, H.G.: Kernel estimation of regression functions (Lect. Notes Math., vol. 757, pp. 23–68) Berlin Heidelberg New York: Springer, 1979

    Google Scholar 

  • Gasser, Th., Müller, H.G.: Estimating regression functions and their derivatives by the kernel method. Scand. J. Stat.11, 171–185 (1984)

    Google Scholar 

  • Hall, P., Heyde, C.C.: Martingale Limit Theory and its Application. New York: Academic Press, 1980

    Google Scholar 

  • Hall, P., Fisher, N.I., Hoffmann, B.: On the nonparametric estimation of covariance functions. Research Report, ANU, 1992

  • Härdle, W.: Applied Nonparametric Regression. Cambridge: University Press, 1990

    Google Scholar 

  • Härdle, W., Marron, J.S.: Random approximations to some measures of accuracy in nonparametric curve estimation. J. Multivariate Anal.20, 91–113 (1986)

    Google Scholar 

  • Ibragimov, I.A., Rozanov, Yu.A.: Gaussian Random Processes. Berlin Heidelberg New York: Springer, 1978

    Google Scholar 

  • Ivanov, A.V., Leonenko, N.N.: Statistical Analysis of Random Fields. Dordrecht: Kluwer, 1989

    Google Scholar 

  • Journel, A.G., Huijbregts, Ch.J.: Mining Geostatistics. London: Academic Press, 1978

    Google Scholar 

  • Kovalchuk, T.V.: On convergence of the distribution of functionals of the correlogram of a homogenous random field with unknown mean. Theor. Probab. Math. Stat.35, 45–53 (1987)

    Google Scholar 

  • Matheron, G.: The Theory of Regionalised Variables. Les Cahiers du Centre de Morphologie Mathematique de Fontainebleau. 5 (1971)

  • Sampson, P.D., Guttorp, P.: Nonparametric representation of nonstationary spatial covariance structure. J. Am. Stat. Assoc.87, 108–119 (1992)

    Google Scholar 

  • Schoenberg, I.J.: Metric spaces and completely monotone functions. Ann. Math.79, 811–841 (1938)

    Google Scholar 

  • Shapiro, A., Botha, J.D.: Variogram fitting with a general class of conditionally nonnegative definite functions. Comput. Stat. Data Anal.11, 87–96 (1991)

    Google Scholar 

  • Silverman, B.W.: Density Estimation for Statistics and Data Analysis. London: Chapman and Hall, 1986

    Google Scholar 

Download references

Author information

Author notes
  1. Peter Hall

    Present address: CSIRO Division of Mathematics and Statistics, Lindfield, NSW, Australia

Authors and Affiliations

  1. Centre for Mathematics and its Applications, Australian National University, 2601, Canberra, ACT, Australia

    Peter Hall & Prakash Patil

Authors
  1. Peter Hall
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Prakash Patil
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hall, P., Patil, P. Properties of nonparametric estimators of autocovariance for stationary random fields. Probab. Th. Rel. Fields 99, 399–424 (1994). https://doi.org/10.1007/BF01199899

Download citation

  • Received: 02 August 1993

  • Revised: 23 December 1993

  • Issue Date: September 1994

  • DOI: https://doi.org/10.1007/BF01199899

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (1991)

  • 62G05
  • 62G20
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature