Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-08T21:14:30.438Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonparametric estimation of the chord length distribution

Part of: Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Martin B. Hansen  and
Erik W. van Zwet
Martin B. Hansen*
Affiliation:
Aalborg University
Erik W. van Zwet*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg Ø, Denmark. Email address: mbh@math.auc.dk
∗∗ Postal address: University of California, Department of Statistics, 367 Evans Hall #3860, Berkeley, CA 94720-3860, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The distribution of the length of a typical chord of a stationary random set is an interesting feature of the set's whole distribution. We give a nonparametric estimator of the chord length distribution and prove its strong consistency. We report on a simulation experiment in which our estimator compared favourably to a reduced sample estimator. Both estimators are illustrated by applying them to an image sample from a yoghurt ferment. We briefly discuss the closely related problem of estimation of the linear contact distribution. We show by a simulation experiment that a transformation of our estimator of the chord length distribution is more efficient than a Kaplan-Meier type estimator of the linear contact distribution.

Keywords

Chord length distributionlinear contact distributionnonparametric maximum likelihood estimationmissing data

MSC classification

Primary: 62M30: Spatial processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 541 - 558
Copyright
Copyright © Applied Probability Trust 2001 

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

Baddeley, A. J. and Gill, R. D. (1997). Kaplan–Meier estimators for interpoint distance distributions of spatial point processes. Ann. Statist. 25, 263–292.CrossRefGoogle Scholar
Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore.Google Scholar
Borgefors, G. (1984). Distance transformations in arbitrary dimensions. Comput. Vision Graph. Image Process. 27, 321–345.Google Scholar
Borgefors, G. (1986). Distance transformations in digital images. Comput. Vision Graph. Image Process. 34, 344–371.Google Scholar
Chiu, S. N. and Stoyan, D. (1998). Estimators of distance distributions for spatial patterns. Statist. Neerlandica 52, 239–246.CrossRefGoogle Scholar
Delfiner, P. (1972). A generalization of the concept of size. J. Microscopy 95, 203–216.CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. R. Statist. Soc. B 39, 1–38.Google Scholar
Gill, R. D. (1994). Lectures on survival analysis. In Lectures on Probability Theory (Lecture Notes Math. 1581), ed. Bernard, P.. Springer, Berlin, pp. 115–241.Google Scholar
Hansen, M. B. (1995). Spatial statistics for network structures in processed milk. , The Royal Veterinary and Agricultural University, Copenhagen.Google Scholar
Hansen, M. B. (1996). Estimation of first contact distribution functions for spatial patterns in S-PLUS. In Proc. COMPSTAT '96, ed. Prat, A.. Physica, Heidelberg, pp. 295–300. Extension of routines available at http://www.math.auc.dk/simmbh/ficodifu/.Google Scholar
Hansen, M. B., Baddeley, A. J. and Gill, R. D. (1996). Kaplan–Meier type estimators for linear contact distributions. Scand. J. Statist. 23, 129–155.Google Scholar
Hansen, M. B., Baddeley, A. J. and Gill, R. D. (1999). First contact distributions for spatial patterns: regularity and estimation. Adv. Appl. Prob. 31, 15–33.Google Scholar
Huang, J. and Wellner, J. A. (1995). Estimation of a monotone density or monotone hazard under random censorship. Scand. J. Statist. 22, 3–33.Google Scholar
Huang, Y. and Zhang, C.-H. (1994). Estimating a monotone density from censored observations. Ann. Statist. 22, 1256–1274.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Statist. 27, 887–906.Google Scholar
Laslett, G. M. (1982a). The survival curve under monotone density constraints with applications to two-dimensional line-segment processes. Biometrika 69, 153–160.Google Scholar
Laslett, G. M. (1982b). Censoring and edge effects in areal and line transect sampling of rock joint traces. Math. Geol. 14, 125–140.CrossRefGoogle Scholar
Le Cam, L. and Yang, G. L. (1988). On the preservation of local asymptotic normality under infromation loss. Ann. Statist. 16, 483–520.Google Scholar
Lum, H., Huang, I. and Mitzner, W. (1990). Morphological evidence for alveolar recruitment during inflation at high transpulmonary pressure. J. Appl. Physiol. 68, 2280–2286.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Oldmixon, E. H., Butler, J. P. and Hoppin, F. G. (1994). Semi-automated measurement of true chord length distributions and moments by video microscopy and image analysis. J. Microscopy 175, 60–69.Google Scholar
Rosenthal, F. S. (1989). Aerosol recovery following breathholding derived from the distribution of chordlengths in pulmonary tissue. J. Aerosol Sci. 20, 267–277.CrossRefGoogle Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology, Vol. 1. Academic Press, London.Google Scholar
Skriver, A. (1995). Characterization of stirred yoghurt by rheology, microscopy and sensory analysis. , Institute for Dairy Research, The Royal Veterinary and Agricultural University, Copenhagen.Google Scholar
Skriver, A., Hansen, M. B. and Qvist, K. B. (1997). Image analysis applied to electron micrographs of stirred youghurt. J. Dairy Res. 64, 135–143.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Van de Geer, S. A. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 21, 14–44.Google Scholar
Van der Laan, M. J. (1995). Efficiency of the NPMLE in the line-segment problem. Scand. J. Statist. 23, 527–550.Google Scholar
Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.Google Scholar
Van Zwet, E. W. (1999). Likelihood devices in spatial statistics. , University of Utrecht.Google Scholar
Wijers, B. J. (1995). Consistent nonparametric estimation for a one-dimensional line segment process observed in an interval. Scand. J. Statist. 22, 335–360.Google Scholar