Estimation variances for parameterized marked Poisson processes and for parameterized Poisson segment processes

Tomáš Mrkvička

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 1, page 109-117
  • ISSN: 0010-2628

Abstract

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A complete and sufficient statistic is found for stationary marked Poisson processes with a parametric distribution of marks. Then this statistic is used to derive the uniformly best unbiased estimator for the length density of a Poisson or Cox segment process with a parametric primary grain distribution. It is the number of segments with reference point within the sampling window divided by the window volume and multiplied by the uniformly best unbiased estimator of the mean segment length.

How to cite

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Mrkvička, Tomáš. "Estimation variances for parameterized marked Poisson processes and for parameterized Poisson segment processes." Commentationes Mathematicae Universitatis Carolinae 45.1 (2004): 109-117. <http://eudml.org/doc/249345>.

@article{Mrkvička2004,
abstract = {A complete and sufficient statistic is found for stationary marked Poisson processes with a parametric distribution of marks. Then this statistic is used to derive the uniformly best unbiased estimator for the length density of a Poisson or Cox segment process with a parametric primary grain distribution. It is the number of segments with reference point within the sampling window divided by the window volume and multiplied by the uniformly best unbiased estimator of the mean segment length.},
author = {Mrkvička, Tomáš},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {complete statistic; compact sets process; intensity estimation; marked point process; Poisson process; random closed sets; Rao-Blackwell Theorem; segment process; spatial statistic; stochastic geometry; sufficient statistic; Poisson segment process; marked point process; sufficient statistic; complete statistic; Rao-Blackwell theorem},
language = {eng},
number = {1},
pages = {109-117},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Estimation variances for parameterized marked Poisson processes and for parameterized Poisson segment processes},
url = {http://eudml.org/doc/249345},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Mrkvička, Tomáš
TI - Estimation variances for parameterized marked Poisson processes and for parameterized Poisson segment processes
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 1
SP - 109
EP - 117
AB - A complete and sufficient statistic is found for stationary marked Poisson processes with a parametric distribution of marks. Then this statistic is used to derive the uniformly best unbiased estimator for the length density of a Poisson or Cox segment process with a parametric primary grain distribution. It is the number of segments with reference point within the sampling window divided by the window volume and multiplied by the uniformly best unbiased estimator of the mean segment length.
LA - eng
KW - complete statistic; compact sets process; intensity estimation; marked point process; Poisson process; random closed sets; Rao-Blackwell Theorem; segment process; spatial statistic; stochastic geometry; sufficient statistic; Poisson segment process; marked point process; sufficient statistic; complete statistic; Rao-Blackwell theorem
UR - http://eudml.org/doc/249345
ER -

References

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  1. Chadoeuf J., Senoussi R., Yao J.F., Parametric estimation of a Boolean segment process with stochastic restoration estimation, J. Comput. Graphical Statistics 9/2 (2000), 390-402. (2000) MR1823807
  2. Daley D.J., Vere-Jones D., An Introduction to the Theory of Point Processes, Springer-Verlag New York (1988). (1988) Zbl0657.60069MR0950166
  3. Lehmann E.L., Theory of Point Estimation, Wadsworth & Brooks California (1991). (1991) Zbl0801.62025MR1143059
  4. Mrkvička T., Estimation Variances for Poisson Process of Compact Sets, (in Czech), Diploma Thesis, Faculty of Mathematics and Physics, Charles University Prague (1999). (1999) 
  5. Mrkvička T., Estimation variances for Poisson Process of Compact Sets, Adv. Appl. Prob. (SGSA) 33 (2001), 765-772. (2001) MR1875778
  6. Stoyan D., Kendall W.S., Mecke J., Stochastic Geometry and its Applications, John Wiley & Sons Chichester (1995). (1995) Zbl0838.60002MR0895588

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