Abstract
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 language | English (US) |
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Pages (from-to) | 99-108 |
Number of pages | 10 |
Journal | STATISTICS AND COMPUTING |
Volume | 14 |
Issue number | 2 |
DOIs | |
State | Published - Apr 2004 |
Externally published | Yes |
Keywords
- 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