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Joint estimators for the specific intrinsic volumes of stationary random sets

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  • Schmidt, Volker
  • Spodarev, Evgueni

Abstract

Stationary random closed sets [Xi] in are considered whose realizations belong to the extended convex ring. A new approach is proposed to joint estimation of the specific intrinsic volumes of [Xi], including the specific Euler-Poincaré characteristic , the specific surface area , and the volume fraction of [Xi]. Nonparametric estimators are constructed, which can be represented by integrals of some stationary random fields. This implies in particular that these estimators are unbiased. Moreover, conditions are derived which ensure that they are mean-square consistent. A consistent estimator for their asymptotic covariance matrix is derived.

Suggested Citation

  • Schmidt, Volker & Spodarev, Evgueni, 2005. "Joint estimators for the specific intrinsic volumes of stationary random sets," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 959-981, June.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:6:p:959-981
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    References listed on IDEAS

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    1. repec:bla:anzsta:v:46:y:2004:i:1:p:41-51 is not listed on IDEAS
    2. S. Böhm & V. Schmidt & L. Heinrich, 2004. "Asymptotic properties of estimators for the volume fractions of jointly stationary random sets," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 58(4), pages 388-406, November.
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    Cited by:

    1. Ursa Pantle & Volker Schmidt & Evgeny Spodarev, 2010. "On the Estimation of Integrated Covariance Functions of Stationary Random Fields," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 47-66, March.
    2. David Dereudre & Frédéric Lavancier & Kateřina Staňková Helisová, 2014. "Estimation of the Intensity Parameter of the Germ-Grain Quermass-Interaction Model when the Number of Germs is not Observed," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(3), pages 809-829, September.
    3. Mrkvicka, T. & Rataj, J., 2008. "On the estimation of intrinsic volume densities of stationary random closed sets," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 213-231, February.

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