Laslett's line segment problem

Abstract

Our problem is to estimate the length distribution of fractures in a rock surface from a geological map. We do not fully observe the fractures because part of the rock surface is covered by vegetation, soil and water. The uncovered region is very irregular and, as a result, we tend to observe several pieces of a single fracture. It is quite impossible to decide from the map if two pieces belong to the same underlying fracture. Under the assumption that the observed pieces are independent, we derive the nonparametric maximum likelihood estimator of the length distribution of the underlying fractures. The assumption is clearly false, but our approach is justified by proving consistency of the estimator without appealing to the independence. We apply our estimator to the geological data.