TY - JOUR
AU - Heinrich, Lothar
TI - LIMIT DISTRIBUTIONS OF SOME STEREOLOGICAL ESTIMATORS IN WICKSELL'S CORPUSCLE PROBLEM
PY - 2011/05/03
Y2 - 2024/10/09
JF - Image Analysis and Stereology
JA - Image Anal Stereol
VL - 26
IS - 2
SE - Original Research Paper
LA -
DO - 10.5566/ias.v26.p63-71
UR - https://doi.org/10.5566/ias.v26.p63-71
SP - 63-71
AB - Suppose that a homogeneous system of spherical particles (d-spheres) with independent identically distributed radii is contained in some opaque d-dimensional body, and one is interested to estimate the common radius distribution. The only information one can get is by making a cross-section of that body with an s-flat (1 ≤ s ≤ d -1) and measuring the radii of the s-spheres and their midpoints. The analytical solution of (the hyper-stereological version of) Wicksell's corpuscle problem is used to construct an empirical radius distribution of the d-spheres. In this paper we study the asymptotic behaviour of this empirical radius distribution for s = d -1 and s = d - 2 under the assumption that the s-dimensional intersection volume becomes unboundedly large and the point process of the midpoints of the d-spheres is Brillinger-mixing. Of course, in stereological practice the only relevant cases are d = 3; s = 2 or s = 1 and d = 2; s = 1. Among others we generalize and extend some results obtained in Franklin (1981) and Groeneboom and Jongbloed (1995) under the Poisson assumption for the special case d = 3; s = 2.
ER -