The Annals of Statistics

Kaplan-Meier estimators of distance distributions for spatial point processes

Adrian Baddeley and Richard D. Gill

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When a spatial point process is observed through a bounded window, edge effects hamper the estimation of characteristics such as the empty space function F, the nearest neighbor distance distribution G and the reduced second-order moment function K. Here we propose and study product-limit type estimators of F, G and K based on the analogy with censored survival data: the distance from a fixed point to the nearest point of the process is right-censored by its distance to the boundary of the window. The resulting estimators have a ratio-unbiasedness property that is standard in spatial statistics. We show that the empty space function F of any stationary point process is absolutely continuous, and so is the product-limit estimator of F. The estimators are strongly consistent when there are independent replications or when the sampling window becomes large. We sketch a CLT for independent replications within a fixed observation window and asymptotic theory for independent replications of sparse Poisson processes. In simulations the new estimators are generally more efficient than the "border method" estimator but (for estimators of K), somewhat less efficient than sophisticated edge corrections.

Article information

Ann. Statist., Volume 25, Number 1 (1997), 263-292.

First available in Project Euclid: 10 October 2002

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 62H11: Directional data; spatial statistics 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Border correction method dilation distance transform edge corrections edge effects empty space statistic erosion functional delta-method influence function $K$-function local knowledge principle nearest-neighbor distance product integration reduced sample estimator reduced second moment measure sparse Poisson asymptotics spatial statistics survival data


Baddeley, Adrian; Gill, Richard D. Kaplan-Meier estimators of distance distributions for spatial point processes. Ann. Statist. 25 (1997), no. 1, 263--292. doi:10.1214/aos/1034276629.

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  • [1] ANDERSEN, N. T., GINE, E., OSSIANDER, M. and ZINN, J. (1988). The central limit theorem and the law of the iterated logarithm for empirical processes under local conditions. Probab. Theory Related Fields 77 271-305.
  • [2] BADDELEY, A. J. (1977). Integrals on a moving manifold and geometrical probability. Adv. in AppL Probab. 9 588-603.
  • [3] BADDELEY, A. J. (1980). A limit theorem for statistics of spatial data. Adv. in AppL Probab. 12 447-461.
  • [4] BADDELEY, A. J. (1993). Stereology and survey sampling theory. Bulletin of the International Statistical Institute 50 435-449.
  • [5] BADDELEY, A, J., MOy EED, R. A., HOWARD, C. V. and BOy DE, A. (1993). Analy sis of a three-dimensional point pattern with replication. J. Roy. Statist. Soc. Ser. C 42 641-668.
  • [6] BARENDREGT, L, G. and ROTTSCHAFER, M, J. (1991). A statistical analysis of spatial point patterns. A case study. Statist. Neerlandica 45 345-363.
  • [7] BORGEFORS, G. (1984). Distance transformations in arbitrary dimensions. Computer Vision, Graphics and Image Processing 27 321-345.
  • [8] BORGEFORS, G. (1986). Distance transformations in digital images. Computer Vision, Graphics and Image Processing 34 344-371.
  • [9] CRESSIE, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York.
  • [10] CROFTON, M. W. (1869). Sur quelques theoremes du calcul integral. C. R. Acad. Sci. Paris 68 1469-1470.
  • [11] DIGGLE, P. J. (1983). Statistical Analy sis of Spatial Point Patterns. Academic Press, London.
  • [12] DIGGLE, P, J. and MATERN, B. (1980). On sampling designs for the estimation of point-event nearest neighbor distributions. Scand. J. Statist. 7 80-84.
  • [13] DOGUWA, S. I. (1989). A comparative study of the edge-corrected kernel-based nearest neighbor density estimators for point processes. J. Statist. Comp. Simulation 33 83-100.
  • [14] DOGUWA, S. I. (1990). On edge-corrected kernel-based pair correlation function estimators for point processes. Biometrical J. 32 95-106.
  • [15] DOGUWA, S, L (1992). On the estimation of the point-object nearest neighbor distribution F(y) for point processes. J. Statist. Comp. Simulation 41 95-107.
  • [16] DOGUWA, S, I. and CHOJI, D. N. (1991). On edge-corrected probability density function estimators for point processes. Biometrical J. 33 623-637.
  • [17] DOGUWA, S. I. and UPTON, G. J. G. (1989). Edge-corrected estimators for the reduced second moment measure of point processes. Biometrical J. 31 563-575.
  • [18] DOGUWA, S. I. and UPTON, G. J. G. (1990). On the estimation of the nearest neighbour distribution, G(£), for point processes, Biometrical J. 32 863-876.
  • [19] FEDERER, H. (1969). Geometric Measure Theory. Springer, Heidelberg.
  • [20] FIKSEL, T. (1988). Edge-corrected density estimators for point processes. Statistics 19 67-75.
  • [21] GILL, R. D. (1989). Nonand semiparametric maximum likelihood estimators and the von Mises method, I (with discussion). Scand. J. Statist. 16 97-128,
  • [22] GILL, R. D. (1994). Lectures on survival analysis. Ecole d'Ete de Probabilites de Saint-Flour 1992. Lecture Notes in Math. 1581. Springer, Berlin.
  • [23] GILL, R. D. and JOHANSEN, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18 1501-1555.
  • [24] HANISCH, K.-H. (1984). Some remarks on estimators of the distribution function of nearest neighbor distance in stationary spatial point patterns. Mathematische Operationsforschung und Statistik-Statistics 15 409-412.
  • [25] HANSEN, M. B., BADDELEY, A. J. and GILL, R. D. (1994). Some regularity properties for first contact distributions. Preprint 890, Mathematics Institute, Univ. Utrecht.
  • [26] HANSEN, M. B., GILL, R. D. and BADDELEY, A. J. (1996). Kaplan-Meier ty pe estimators for linear contact distributions. Scand, J. Statist. 23 129-155.
  • [27] HEINRICH, L. (1988). Asy mptotic Gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes. Statistics 19 87-106.
  • [28] JOLIVET, E. (1980). Central limit theorem and convergence of empirical processes for stationary point processes. In Point Processes and Queueing Problems (P. Bastfai and J. Tomko, eds.) 117-161. North-Holland, Amsterdam.
  • [29] KAPLAN, E, L. and MEIER, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457-481.
  • [30] LASLETT, G. M. (1982). Censoring and edge effects in areal and line transect sampling of rock joint traces. Math. Geol. 14 125-140.
  • [31] LASLETT, G. M. (1982). The survival curve under monotone density constraints with applications to two-dimensional line segment processes. Biometrika 69 153-160.
  • [32] LOTWICK, H. W. (1981). Spatial stochastic point processes. Ph.D. thesis, Univ. Bath.
  • [33] MATHERON, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
  • [34] MILES, R. E. (1974). On the elimination of edge-effects in planar sampling. In Stochastic Geometry: A Tribute to the Memory of Rollo Davidson (E. F. Harding and D. G. Kendall, eds.) 228-247. Wiley, New York.
  • [35] OHSER, J. (1983). On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik-Statistics 14 63-71.
  • [36] RIPLEY, B. D. (1977). Modelling spatial patterns (with discussion). J. Roy. Statist. Soc. Ser. B 39 172-212.
  • [37] RIPLEY, B. D. (1981). Spatial Statistics. Wiley, New York.
  • [38] RIPLEY, B. D. (1988). Statistical Inference for Spatial Processes. Cambridge Univ. Press.
  • [39] ROSENFELD, A. and PFALZ, J. L. (1966). Sequential operations in digital picture processing. J. Assoc. Comput. Mach. 13 471.
  • [40] ROSENFELD, A. and PFALZ, J. L. (1968). Distance functions on digital pictures. Pattern Recognition 1 33-61.
  • [41] SANTAL6, L. A. (1976). Integral Geometry and Geometric Probability. Ency clopedia of Mathematics and Its Applications 1. Addison-Wesley, Reading, MA.
  • [42] SERRA, J. (1982). Image Analy sis and Mathematical Morphology. Academic Press, London.
  • [43] STEIN, M. L. (1991). A new class of estimators for the reduced second moment measure of point processes. Biometrika 78 281-286.
  • [44] STEIN, M. L. (1993). Asy mptotically optimal estimation for the reduced second moment measure of point processes. Biometrika 80 443-449.
  • [45] STEIN, M. L. (1995). An approach to asy mptotic inference for spatial point processes. Statistica Sinica 5 221-234.
  • [46] STOy AN, D., BERTRAM, U. and WENDROCK, H. (1993). Estimation variances for estimators of product densities and pair correlation functions of planar point processes. Ann. Inst. Statist. Math. 45 211-221.
  • [47] STOy AN, D., KENDALL, W. S. and MECKE, J. (1987). Stochastic Geometry and Its Applications. Wiley, Chichester.
  • [48] TEN KATE, T. K., VAN BALEN, R, SMEULDERS, A. W. M., GROEN, F. C. A. and DEN BOER, G. A.
  • (1990). SCILAIM: a multi-level interactive image processing environment. Pattern Recognition Letters 11 429-441.
  • [49] VAN DER VAART, A. and WELLNER, J. A. (1993). Weak Convergence and Empirical Processes.
  • IMS, Hay ward, CA.
  • [50] WlJERS, B. J. (1991). Consistent non-parametric estimation for a one-dimensional line segment process observed in an interval. Preprint 683, Dept. Mathematics, Univ. Utrecht.