Mathematical Geosciences

, Volume 43, Issue 3, pp 363–377 | Cite as

Combining Robustness with Efficiency in the Estimation of the Variogram

  • Hilário MirandaEmail author
  • Manuela Souto de Miranda


In the present paper, we propose a new method for the estimation of the variogram, which combines robustness with efficiency under intrinsic stationary geostatistical processes. The method starts by using a robust estimator to obtain discrete estimates of the variogram and control atypical observations that may exist. When the number of points used in the fit of a model is the same as the number of parameters, ordinary least squares and generalized least squares are asymptotically equivalent. Therefore, the next step is to fit the variogram by ordinary least squares, using just a few discrete estimates. The procedure is then repeated several times with different subsets of points and this produces a sequence of variogram estimates. The final estimate is the median of the multiple estimates of the variogram parameters. The suggested estimator will be called multiple variograms estimator. This procedure assures a global robust estimator, which is more efficient than other robust proposals. Under the assumed dependence structure, we prove that the multiple variograms estimator is consistent and asymptotically normally distributed. A simulation study confirms that the new method has several advantages when compared with other current methods.


Spatial statistics Multiple variograms estimator Robust estimator Bounded influence function Breakdown point 


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  1. Bassoi L (1994) Nitrato no solo e acumulo de N pelo milho (Zea mays L) fertirrigado. PhD Thesis, University of São Paulo, Brazil Google Scholar
  2. Cressie N (1985) Fitting variogram models by weighted least squares. J Int Assoc Math Geol 17:693–702 CrossRefGoogle Scholar
  3. Genton M (1998a) Highly robust variogram estimation. Math Geol 30(2):213–221 CrossRefGoogle Scholar
  4. Genton M (1998b) Variogram fitting by generalized least squares using an explicit formula for the covariance structure. Math Geol 30(4):323–345 CrossRefGoogle Scholar
  5. Genton M (1998c) Spatial breakdown point of variogram estimators. Math Geol 30(7):853–871 CrossRefGoogle Scholar
  6. Hampel F, Ronchetti E, Rousseeuw P, Stabel W (1986) Robust statistics: the approach based on influence functions. Wiley, New York Google Scholar
  7. Journel A, Huijbregts C (1978) Mining geostatistics. Academic Press, London Google Scholar
  8. Koenker R (2005) Quantile regression. Cambridge University Press, Cambridge CrossRefGoogle Scholar
  9. Lahiri S, 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 CrossRefGoogle Scholar
  10. Maronna R, Martin R, Yohai V (2006) Robust statistics—theory and methods. Wiley, London CrossRefGoogle Scholar
  11. Matheron G (1962) Traite de geostatistique appliquee, vol I. Memoires du bureau de recherches geologiques et minieres, vol 14. Editions Technip, Paris Google Scholar
  12. Mizera I, Wellner J (1998) Necessary and sufficient conditions for weak consistency of the median of independent but not identically distributed random variables. Ann Stat 26(2):672–691 CrossRefGoogle Scholar
  13. R Development Core Team (2008) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna Google Scholar
  14. Ribeiro P Jr., Diggle P (2001) geoR: A package for geostatistical analysis. R-News 1(2):15–18 Google Scholar
  15. Rousseeuw P, Croux C (1993) Alternatives to the median absolute deviation. J Am Stat Assoc 88(424):1273–1283 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2010

Authors and Affiliations

  1. 1.Portucalense University, Infante D. HenriquePortoPortugal
  2. 2.Department of MathematicsUniversity of AveiroAveiroPortugal

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