Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T19:31:48.125Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of the intensity of stationary flat processes

Part of: Nonparametric inference General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Katja Schladitz
Katja Schladitz*
Affiliation:
ITWM Kaiserslautern
*
∗ Postal address: ITWM (Institut für Techno- und Wirtschaftsmathematik), Erwin-Schrödinger-Straße, 67663 Kaiserslautern, Germany. Email address: schlad@itwm.uni-kl.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The intensity of a stationary process of k-dimensional affine subspaces (k-flats) of ℝd with directional distribution from a given family R is estimated by observing the process in a compact window. To this end we introduce a type of unbiased estimator (the R-estimator) using the available information about the directional distribution.

Special cases are estimators for the intensity of stationary k-flat processes (1) with known directional distribution, (2) with directional distribution invariant with respect to a subgroup of the group of rotations in ℝd and (3) with unknown directional distribution.

We give sufficient conditions for the R-estimator to be the uniformly best unbiased estimator for the intensity of stationary Poisson k-flat processes with directional distribution in R. Equivalent statements for certain types of stationary Cox flat processes can be deduced directly from the results in the Poisson case.

Moreover, we consider stationary ergodic flat processes with directional distribution in R and general stationary flat processes with unknown directional distribution, all with a non-degeneracy property. In both cases our estimator turns out to be the uniformly best unbiased estimator from a restricted set of estimators. The result for general stationary flat processes is proved with the help of a factorization result for the second factorial moment measure.

Keywords

Stationary hyperplane processstationary line processPoisson flat processesintensityunbiased estimatorDavidson's conjecturesecond factorial moment measuresufficiencysymmetric completeness

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 62G05: Estimation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 114 - 139
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ambartzumian, R. V. (1976). A note on pseudo-metrics on the plane. Z. Wahrscheinlichkeitsth. 37, 145–155.CrossRefGoogle Scholar
[2] Ambartzumian, R. V. (1982). Combinatorial Integral Geometry. John Wiley, New York. [With an Appendix by A. Baddeley.].Google Scholar
[3] Ambartzumian, R. V. (1990). Factorization Calculus and Geometric Probability. Cambridge University Press.Google Scholar
[4] Baddeley, A. J. and Cruz-Orive, L. M. (1995). The Rao–Blackwell theorem in stereology and some counterexamples. Adv. Appl. Prob. 27, 2–19.Google Scholar
[5] Barndorff-Nielsen, O., Blæsild, P. and Eriksen, P. (1989). Decomposition and Invariance of Measures, and Statistical Transformation Models (Lecture Notes in Stat. 58). Springer, Berlin.Google Scholar
[6] Cruz-Orive, L. M. (1993). Systematic sampling on the semicircle. Lecture at the 6th Workshop on Stochastic Geometry, Stereology and Image Analysis, Valencia.Google Scholar
[7] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes, Springer, Berlin.Google Scholar
[8] Davidson, R. (1974). Construction of line processes: second-order properties. In Stochastic Geometry, eds Harding, E. F. and Kendall, D. G.. John Wiley, Chichester, pp. 55–75.Google Scholar
[9] Fellous, A. and Granara, J. (1981). Statistique des processus de Poisson stationnaires de sous-variétés linéaires affines. Adv. Appl. Prob. 13, 84–92.Google Scholar
[10] Fellous, A., Granara, J. and Krickeberg, K. (1978). Statistics of stationary oriented line poisson processes in the plane. In Geometric Probability and Biological Structures: {Buffon's} 200th anniversary, eds Miles, R. E. and Serra, J., No. 23 in LNBM. Springer, Berlin, pp. 295–299.Google Scholar
[11] Kallenberg, O. (1976). On the structure of stationary flat processes. Z. Wahr-scheinlichkeitsth. 37, 157–174.Google Scholar
[12] Kallenberg, O. (1977). A counterexample to {R. Davidson's} conjecture on line processes. Math. Proc. Camb. Phil. Soc. 82, 301–307.Google Scholar
[13] Krickeberg, K. (1974). Invariance properties of the correlation measure of line-processes. In Stochastic Geometry, eds Harding, E. F. and Kendall, D. G., John Wiley, Chichester, pp. 76–88.Google Scholar
[14] Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. John Wiley, Chichester.Google Scholar
[15] Mecke, J. (1967). Stationäre zufällige Maäe auf lokal-kompak-ten Abel-schen Gruppen. Z. Wahr-scheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
[16] Mecke, J. (1979). An explicit description of {Kallenberg's} lattice type process. Math. Nachr. 89, 185–195.Google Scholar
[17] Mecke, J., Schneider, R. G., Stoyan, D. and Weil, W. R. R. (1990). Stochastische Geometrie. Birkhäuser, Basel.Google Scholar
[18] Neumann, P. M., Stoy, G. A. and Thompson, E. C. (1994). Groups and Geometry. Oxford University Press.Google Scholar
[19] Neveu, J. (1972). Martingales à temps discret. Masson, Paris.Google Scholar
[20] Ohser, J. (1990). Grundlagen und praktische Mäglichkeiten der Cha-rak-te-ri-sie-rung struktureller Inhomo-genitäten von Werkstoffen. Dr. sc. , Bergakademie Freiberg, Freiberg, germany.Google Scholar
[21] Pfanzagl, J. (1994). Parametric Statistical Theory. de Gruyter, Berlin.Google Scholar
[22] Schladitz, K. (1994). Unbiased estimators for the length intensity of stationary planar line processes. {Forschungsergebnisse der Mathematischen Fakultät}, Friedrich-Schiller-Universität Jena.Google Scholar
[23] Schladitz, K. (1996). Estimation of the intensity of stationary flat processes. , Friedrich-Schiller-Universität Jena.CrossRefGoogle Scholar
[24] Schneider, R. and Weil, W. (1986). Translative and kinematic integral formulae for curvature measures. Math. Nachr. 129, 67–80.Google Scholar
[25] Stoyan, D., Kendall, W. S. and Mecke, J. (1989). Stochastic Geometry and its Applications, 2nd edn, Akademie, Berlin.Google Scholar
[26] Sukiasian, G. S. (1987). Randomizable point systems. Acta Appl. Math. 9, 83–95.Google Scholar
[27] Weil, W. (1987). Point processes of cylinders, particles and flats. Acta Appl. Math. 9, 103–136.Google Scholar