Set-Valued Estimators for Mean Bodies Related to Boolean Models

References

• Cressie , N. A. C. and Laslett , G. M. 1987 . Random set theory and problems of modeling . SIAM Review , 29 : 557 – 574 .
   Web of Science ®Google Scholar
• Cressie , N. A. C. 1991 . Statistics for Spatial Data , New York : Wiley . etc
   Google Scholar
• Hall , P. 1988 . Indroduction to the Theory of Coverage Processes , New York : Wiley . etc
   Google Scholar
• Heinrich , L. 1992 . On existence and mixing properties of germ-grain models . Statistics , 23 : 271 – 286 .
   Google Scholar
• Heinrich , L. 1993 . Asymptotic properties of minimum contrast estimators for parameters of Boolean models . Metrika , 31 : 349 – 360 .
   Google Scholar
• Lantuejoul , C. and Schmitt , M. 1991 . Use of two new formulae to estimate the Poisson intensity of a Boolean model . Treizième Colloque GRETSI — Juan-Les-Pins du 16 au 20 Septembre 1991 , : 1045 – 1048 .
   Google Scholar
• Leichtweiss , K. 1980 . Konvexe Mengen , Berlin : VEB Deutscher Verlag der Wissenschaften .
   Google Scholar
• Mase , S. 1982 . Asymptotic properties of stereological estimators of volume fraction for stationary random sets . J. Appl. Prob. , 19 : 111 – 126 .
   Web of Science ®Google Scholar
• Matheron , G. 1975 . Random Sets and Integral Geometry , New York : Wiley .
   Google Scholar
• Mecke , J. and Stoyan , D. 1980 . Formulas for stationary planar fibre processes I - general theory . Math. Operationsf. Statist., ser. statist. , 12 : 267 – 279 .
   Google Scholar
• Molchanov , I. S. 1988 . Convergence of the empirical accompanying functional of stationary random sets . Theory Probab. Math. Statist. , 39 : 107 – 109 .
   Google Scholar
• Molchanov , I. S. 1990 . Estimation of the size distribution of spherical grains in the Boolean model . Biometrical J. , 32 : 877 – 886 .
   Web of Science ®Google Scholar
• Molchanov , I. S. 1991 . A consistent estimate of parameters of Boolean models of random closed sets . Theory Probab. Appl. , 36 : 600 – 607 .
   Web of Science ®Google Scholar
• Molchanov , I. S. 1994 . On statistical analysis of Boolean models with non-random grains . Scand. J. Statist. , 21 : 73 – 82 .
   Web of Science ®Google Scholar
• Molchanov , I. S. 1995 . Statistics of the Boolean model: From the estimation of means to the estimation of distributions . Adv. in Appl. Probab. , 27 : 63 – 86 .
   Web of Science ®Google Scholar
• Molchanov , I. S. and Stoyan , D. 1994a . Directional analysis of fibre processes related to Boolean models . Metrika , 41 : 183 – 199 .
   Google Scholar
• Molchanov , I. S. and Stoyan , D. 1994b . Asymptotic properties of estimators for parameters of the Boolean model . Adv. in Appl. Probab. , 26 : 301 – 323 .
   Web of Science ®Google Scholar
• Rataj , J. and Saxl , I. 1989 . Analysis of planar anisotropy by means of the Steiner compact . J. Appl. Probab. , 26 : 490 – 502 .
   Web of Science ®Google Scholar
• Rataj , J. and Saxl , I. 1992 . Estimation of direction distribution of a planar fibre system . Acta Sterologica. , 11 : 631 – 636 . Suppl.l
   Google Scholar
• Ripley , B. D. 1986 . Statistics, images and pattern recognition . Canadian J. Statist. , 14 : 83 – 102 .
   Web of Science ®Google Scholar
• Schmitt , M. 1991 . Estimation of the density in a stationary Boolean model . J. Appl. Probab. , 28 : 702 – 708 .
   Web of Science ®Google Scholar
• Schröder , M. 1992 . Schätzer für Boolesche Modelle im R² und R³ , Diplomarbeit : Universität Karlsruhe .
   Google Scholar
• Serra , J. 1982 . Image Analysis and Mathematical Morphology , London : Academic Press .
   Google Scholar
• Stoyan , D. , Kendall , W. S. and Mecke , J. 1987 . Stochastic Geometry and its Applications , Berlin : Akademie - Verlag . Wiley, Chichester
   Google Scholar
• Stoyan , D. and Stoyan , H. 1994 . Fractals, Random Shapes and Point Fields. Methods of Geometrical Statistics. , Wiley, Chichester . etc
   Google Scholar
• Vitale , R. 1988 . An alternate formulation of mean value for random geometric figures . J. Microscopy , 151 : 197 – 204 .
   Web of Science ®Google Scholar
• Weil , W. 1988 . Expectation formulas and isoperimetric properties for non-isotropic Boolean models . J. Microscopy , 151 : 235 – 245 .
   Web of Science ®Google Scholar
• Weil , W. 1993 . The determination of shape and mean shape from sections and projections . Acta Stereologica , 12/2 : 73 – 84 .
   Google Scholar
• Weil , W. 1994 . Support functions on the convex ring in the plane and support densities for random sets and point processes . Rend. Cire. Mat. Palermo , 35 : 323 – 344 .
   Google Scholar
• Weil , W. 1995 . The estimation of mean shape and mean particle number in overlapping particle systems in the plane . Adv. in Appl. Probab. , 27 : 102 – 119 .
   Web of Science ®Google Scholar
• Yadin , M. and Zacks , S. 1985 . The visibility of stationary and moving targets in the plane subject to a Poisson field of shadowing elements . J. Appl. Prob. , 22 : 776 – 786 .
   Web of Science ®Google Scholar