University of Illinois - Chicago - College Of Medicine
The linear representation of order statistics is a random permutation matrix which can be applied to obtain the usual covariance structure of ranks and other induced order statistics. In this talk, the algebraic structure of the standard case will be identified and extended to the ordering of observations indexed by circular, uniformly spaced, coordinates. These data are characteristic, for example, of corneal curvature maps used to assess regular astigmatism in the optics of the human eye. To obtain the covariance structure among the angular displacements in the coordinates (circular ranks), induced by the ordered observations, the cyclic group mean and mean conjugate are derived. Assumptions of cyclic permutation invariance for the underlying probability models will be considered. The covariance structure resulting from different symmetries (e.g., those of the regular polygons) will also be discussed. The treatment of repeated data (ties), it will be shown, is also a direct consequence of the proposed linear representation.