Test

, Volume 5, Issue 1, pp 61–76 | Cite as

Asymptotic expansions for statistics computed from spatial data

  • P. García-Soidán
Article

Summary

The Edgeworth expansions for dependent data are generalized to the context of spatial patterns, with the aim of obtaining asymptotic expansions which approximate the distribution of statistics computed from spatial data, generated by a weakly dependent coverage process. In particular, the case of estimating the expected proportion (its porosity) of a region that is not covered by the process is treated in detail and explicit formulae are given in the context of a Boolean model, assuming that the random sets generating the model are essentially bounded and satisfy a version of Cramér’s condition.

Keywords

Boolean Model Coverage Process Cramér’s Condition Edgeworth Expansion Porosity Vacancy 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bhattacharya, R. N. and Ranga Rao, R. (1976).Normal Approximations and Asymptotic Expansions. New York: Wiley.Google Scholar
  2. Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansions.Ann. Statist. 6, 434–451.MathSciNetGoogle Scholar
  3. Götze, F. and Hipp, C. (1983). Asymptotic expansions for sums of weakly dependent random vectors.Z. Wahrscheinlichkeitstheorie verw. Geb. 64, 211–239.CrossRefGoogle Scholar
  4. Götze, F. and Hipp, C. (1989). Asymptotic expansions for potential functions of i.i.d. random fields.Prob. Theory and Related Fields 82, 349–370.Google Scholar
  5. Hall, P. (1988).Introduction to the Theory of Coverage Processes. New York: Wiley.MATHGoogle Scholar
  6. Hall, P. (1992).The Bootstrap and the Edgeworth Expansion. New York: Springer-Verlag.Google Scholar
  7. Heinrich, L. (1987). Asymptotic expansions in the CLT for a special class ofm-dependent random fields I—Lattice case.Math. Nachr. 134, 83–106.MathSciNetGoogle Scholar
  8. Heinrich, L. (1990). Asymptotic expansions in the CLT for a special class ofm-dependent random fields II—Lattice case.Math. Nachr. 145, 309–327.MathSciNetGoogle Scholar
  9. Heinrich, L. (1993). Asymptotic properties of minimum contrast estimators for parameters of Boolean models.Metrika 40, 67–94.MathSciNetGoogle Scholar
  10. Heinrich, L. and Molchanov, I. S. (1995). Central limit theorem for a class of random measures associated with germ-grain models.CWI Reports, Dept. of Operations Research, Statistics and System theory, Amsterdam.Google Scholar

Copyright information

© SEIO 1996

Authors and Affiliations

  • P. García-Soidán
    • 1
  1. 1.Departamento de Estadística e Investigación OperativaUniversidad de VigoVigoSpain

Personalised recommendations