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This paper compares estimators of u = |P2 — P1| based on partial knowledge about points P1, P2 which are chosen independently from the same probability distribution in the plane. Typical estimators are the conditional expectations E(u | r1, r2), E(u | s), E(u | rM), E(u | θ) where ri is the distance from Pi to the origin 0, s = r1 + r2, rM = Max{r1, r2}, θ = ∠ P10P2. These expectations are the least squares estimators, given the conditioned variables. Simpler linear estimators are also considered.

Let Dj be a domain in nj,-dimensional Euclidean space, for j = 1, …, k. Suppose that for each j = 1,…, k, Nj points are chosen independently at random in Dj. A theorem, which is an extension of a theorem of Crofton, is proved about the expected value of functions of the points.

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Estimators for distances

A more general form of a theorem of Crofton

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2 results

Estimators for distances

A more general form of a theorem of Crofton