Probability Theory and Related Fields

, Volume 99, Issue 3, pp 399–424 | Cite as

Properties of nonparametric estimators of autocovariance for stationary random fields

  • Peter Hall
  • Prakash Patil


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.

Mathematics Subject Classification (1991)

62G05 62G20 


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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Peter Hall
    • 1
  • Prakash Patil
    • 1
  1. 1.Centre for Mathematics and its ApplicationsAustralian National UniversityCanberraAustralia

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