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, Volume 16, Issue 3, pp 479–503 | Cite as

Asymptotic normality of the Nadaraya–Watson semivariogram estimators

Original Paper

Abstract

In this work, the Nadaraya–Watson semivariogram estimation is considered for both the isotropic and the anisotropic settings. Several properties of these estimators are analyzed and, particularly, their asymptotic normality is established in terms of unknown characteristics of the random process. The latter provides a theoretical procedure for construction of confidence intervals for the semivariogram via the normal quantiles, which in practice must be appropriately estimated. A numerical study is included to illustrate the performance of the Nadaraya–Watson estimation when used to obtain confidence intervals.

Keywords

Asymptotic normality Intrinsic stationarity Isotropy Kernel Random process Variogram 

Mathematics Subject Classification (2000)

62G05 62G10 

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References

  1. Baczkowsky AJ, Mardia KV (1987) Approximate lognormality of the sample semi-variogram under a Gaussian process. Commun Stat Simul Comput 16:571–585 CrossRefGoogle Scholar
  2. Cressie N (1993) Statistics for spatial data. Wiley, New York Google Scholar
  3. Cressie N, Hawkins DM (1980) Robust estimation of the variogram, I. Math Geol 12:115–125 CrossRefMathSciNetGoogle Scholar
  4. Davis BM, Borgman LE (1982) A note on the asymptotic distribution of the sample variogram. Math Geol 14:189–193 CrossRefMathSciNetGoogle Scholar
  5. García-Soidán PH, Febrero-Bande M, González Manteiga W (2004) Nonparametric kernel estimation of an isotropic semivariogram. J Stat Plan Inference 121:65–92 MATHCrossRefGoogle Scholar
  6. García-Soidán PH, González-Manteiga W, Febrero-Bande M (2003) Local linear regression estimation of the variogram. Stat Probab Lett 64:169–179. MATHCrossRefGoogle Scholar
  7. Genton M (1998) Highly robust variogram estimation. Math Geol 30:213–221 MATHCrossRefMathSciNetGoogle Scholar
  8. Hall P, Fisher NI, Hoffmann B (1994) On the nonparametric estimation of covariance functions. Ann Stat 22:2115–2134 MATHMathSciNetGoogle Scholar
  9. Hall P, Patil P (1994) Properties of nonparametric estimators of autocovariance for stationary random fields. Probab Theory Relat Fields 99:399–424 MATHCrossRefMathSciNetGoogle Scholar
  10. Lahiri SN, Lee Y, Cressie N (2002). On asymptotic distribution and asymptotic efficiency of least squares estimators of spatial variogram parameters. J Stat Plan Inference 103:65–85 MATHCrossRefMathSciNetGoogle Scholar
  11. Matheron G (1963) Principles of geostatistics. Econ Geol 58:1246–1266 CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2007

Authors and Affiliations

  1. 1.Facultad de Ciencias Sociales y de la ComunicaciónUniversidad de VigoPontevedraSpain

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