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Alignments in two-dimensional random sets of points

Published online by Cambridge University Press:  01 July 2016

David G. Kendall  and
Wilfrid S. Kendall
David G. Kendall*
Affiliation:
University of Cambridge
Wilfrid S. Kendall*
Affiliation:
University of Hull
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, U.K.
∗∗Postal address: Department of Mathematical Statistics, The University, Hull HU6 7RX, U.K.
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Abstract

Let n points in the plane be generated by some specified random mechanism and suppose that N(∊) of the resulting triads form triangles with largest angle ≧ π – ∊. The main object of the paper is to obtain asymptotic formulae for and Var (N(∊)) when ∊ ↓ 0, and to solve the associated data-analytic problem of testing whether an empirical set of n points should be considered to contain too many such ∊-blunt triads in the situation where the generating mechanism is unknown and where all that can be said about the tolerance ∊ is that it must be allowed to take values anywhere in a given interval (T0, T1) (0 < T0 < T1). This problem is solved by the introduction of a plot to be called the pontogram and by the introduction of simulation-based significance tests constructed by random lateral perturbations of the data.

Keywords

ALIGNMENTSBLUNTNESSBROADBENT COEFFICIENTBROWNIAN BRIDGECOLLINEARITY CONSTANTSGAUSSIAN TRIANGLESLATERAL PERTURBATIONSMULTIPLE COLLINEARITIESORNSTEIN–UHLENBECK PROCESSOVER-UNROUNDINGPONTOGRAMSECOND-ORDER TOLERANCESSHAPE-DENSITYSHAPE-FUNCTIONSHAPE-MANIFOLDSPHERICAL BLACKBOARD
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 2 , June 1980 , pp. 380 - 424
Copyright
Copyright © Applied Probability Trust 1980 

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