Kaplan-Meier estimators of distance distributions for spatial point processes

Annals of Statistics

Volumn 25, Issue 1, 1997, Pages 263-292

  Baddeley, Adrian ^a   Gill, Richard D ^b  

^a UNIVERSITY OF WESTERN AUSTRALIA   (Australia)

^b UTRECHT UNIVERSITY   (Netherlands)

Author keywords

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

Indexed keywords

EID: 0031511026     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1034276629     Document Type: Article

References (51)

1. ANDERSEN, N. T., GINÉ, 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., MOYEED, R. A., HOWARD, C. V. and BOYDE, A. (1993). Analysis of a three-dimensional point pattern with replication. J. Roy. Statist. Soc. Ser. C 42 641-668.
6. BARENDREGT, L. G. and ROTTSCHÄFER, 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 théorèmes du calcul intégral. C. R. Acad. Sci. Paris 68 1469-1470.
11. DIGGLE, P. J. (1983). Statistical Analysis of Spatial Point Patterns. Academic Press, London.
12. DIGGLE, P. J. and MATÉRN, 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. I. (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(t), 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). Non-and 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'Eté de Probabilités 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 type estimators for linear contact distributions. Scand. J. Statist. 23 129-155.
27. HEINRICH, L. (1988). Asymptotic 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. SANTALÓ, L. A. (1976). Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and Its Applications 1. Addison-Wesley, Reading, MA.
42. SERRA, J. (1982). Image Analysis 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). Asymptotically optimal estimation for the reduced second moment measure of point processes. Biometrika 80 443-449.
45. STEIN, M. L. (1995). An approach to asymptotic inference for spatial point processes. Statistica Sinica 5 221-234.
46. STOYAN, 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. STOYAN, 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, Hayward, CA.
50. WIJERS, B. J. (1991). Consistent non-parametric estimation for a one-dimensional line segment process observed in an interval. Preprint 683, Dept. Mathematics, Univ. Utrecht.
51. ZIMMERMAN, D. (1991). Censored distance-based intensity estimation of spatial point processes. Biometrika 78 287-294.