• Zbynek Pawlas Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University
  • Marketa Zikmundova Department of Mathematics, Faculty of Chemical Engineering, University of Chemistry and Technology, Prague




Horvitz-Thompson estimator, Kaplan-Meier estimator, line segment process, nonparametric maximum likelihood estimator, reduced-sample estimator, SRE algorithm


We study nonparametric estimation of the length distribution for stationary line segment processes in the d-dimensional Euclidean space. Several methods have been proposed in the literature. We review different approaches (Horvitz-Thompson type estimator, reduced-sample estimator, Kaplan-Meier estimator, nonparametric maximum likelihood estimator, stochastic restoration estimation) and compare the finite sample behaviour by means of a simulation study for stationary line segment processes in 2D and 3D. Several data generating processes (Poisson point process, Matérn cluster process and Matérn hard-core process II) are considered with both independent and dependent segments. Our finite sample comparison reveals that the nonparametric likelihood estimator provides the most preferable method which works reasonably also if its assumptions are not satisfied. 


Baddeley AJ (1999). Spatial sampling and censoring. In: Barndorff-Nielsen OE, Kendall WS, van Lieshout MNM, eds. Stochastic Geometry: Likelihood and Computation. London: Chapman and Hall, 37--78.

Baddeley AJ, Gill RD (1997). Kaplan--Meier estimators of distance distributions for spatial point processes. Ann Statist 25:263--92.

Baddeley AJ, Turner R (2005). Spatstat: an R package for analyzing spatial point patterns. J Stat Softw 12:1--42.

Chadoeuf J, Senoussi R, Yao JF (2000). Parametric estimation of a Boolean segment process with stochastic restoration estimation. J Comput Graph Statist 9:390--402.

Heinrich L, Pawlas Z (2008). Weak and strong convergence of empirical distribution functions from germ-grain processes. Statistics 42:49--65.

Illian J, Penttinen A, Stoyan H, Stoyan D (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Chichester: John Wiley & Sons.

Kaplan EL, Meier P (1958). Nonparametric estimation from incomplete observations. J Amer Statist Assoc 53:457--81.

Kuhlmann M, Redenbach C (2015). Estimation of fibre length distributions from fibre endpoints. Scand J Statist 42:1010--22.

Laslett GM (1982a). Censoring and edge effects in areal and line transect sampling of rock joint traces. Math Geol 14:125--40.

Laslett GM (1982b). The survival curve under monotone density constraints with applications to two-dimensional line segment processes. Biometrika 69:153--60.

Pawlas Z (2006). Estimation of the distribution function in germ-grain models. In: Huskova M, Janzura M, eds. Proceedings Prague Stochastics 2006. Prague: Matfyzpress, 579--89.

Pawlas Z, Honzl O (2010). Comparison of length-intensity estimators for segment processes. Statist Probab Lett 80:825--33.

R Core Team (2018). R: A language and environment for statistical

computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.

Schneider R, Weil W (2008). Stochastic and Integral Geometry. Berlin: Springer-Verlag.

Stephens MA (1992). Introduction to Kolmogorov (1933) On the empirical determination of a distribution. In: Kotz S, Johnson NL, eds. Breakthroughs in Statistics: Methodology and Distribution. New York: Springer-Verlag, 93--105.

Svensson I, Sjoestedt-de Luna S, Bondesson L (2006). Estimation of wood fibre length distributions from censored data through an EM algorithm. Scand J Statist 33:503--22.

van Zwet EW (2004). Laslett's line segment problem. Bernoulli 10:377--96.

Wijers BJ (1997). Nonparametric estimation for a~windowed line-segment process. Amsterdam: Stichting Mathematisch Centrum.




How to Cite

Pawlas, Z., & Zikmundova, M. (2019). A COMPARISON OF NONPARAMETRIC ESTIMATORS FOR LENGTH DISTRIBUTION IN LINE SEGMENT PROCESSES. Image Analysis and Stereology, 38(2), 121–130. https://doi.org/10.5566/ias.1889



Original Research Paper