On the discretization of nonparametric isotropic covariogram estimators

David J. Gorsich*, Marc G. Genton

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


In this article, we describe the discretization of nonparametric covariogram estimators for isotropic stationary stochastic processes. The use of nonparametric estimators is important to avoid the difficulties in selecting a parametric model. The key property the isotropic covariogram must satisfy is to be positive definite and thus have the form characterized by Yaglom's representation of Bochner's theorem. We present an optimal discretization of the latter in the sense that the resulting nonparametric covariogram estimators are guaranteed to be smooth and positive definite in the continuum. This provides an answer to an issue raised by Hall, Fisher and Hoffmann (1994). Furthermore, from a practical viewpoint, our result is important because a nonlinear constrained algorithm can sometimes be avoided and the solution can be found by least squares. Some numerical results are presented for illustration.

Original languageEnglish (US)
Pages (from-to)99-108
Number of pages10
Issue number2
StatePublished - Apr 2004
Externally publishedYes


  • Bochner's theorem
  • Fourier-Bessel series
  • Nonnegative least squares
  • Positive definiteness
  • Spatial prediction

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics


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