Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Ghosh, Jayanta Kumar (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

In many natural and physical sciences the obser- vations are in the form of directions in 2- or 3-dimensional Euclidean space or rotations of such a space. In the former case, it is tustomary to represent the observations by points on the unit ball in R2 or R3, or more.generally, Rp. In the latter case, the observations are represented by orthogonal matrices having determinant +1, or more generally, by nxp (ngp) matrices A satisfying AAt=In,i.e., by elements of Stiefel manifold. Examples .of observations on directions in 2-dimensional space include those on wind direction or on flight direction of birds, Similarly, with directions in 3-dimensional space one comes across obser-vations on arrival directions of showers of cosmic rays or on facing directions of conically folded planes. Other interesting examples of observations on directions in 2- or 3-dimensional spaces may be found in Mardia (1972), Batschelet (1981), and Fisher et al. (1987). On the other hand, examples of observations from Stiefel manifold include the specifications of elliptical cometary orbits by their perihelion and normal directions (Mardia (1975a), Jupp and Mardia (1979)) and similar orbits; in vectorcardio- graphy (Downs (1972), Prentice (1986)).However, it should be mentioned that the study of directional statistics gives rise to statistical problems which do not fit into the usual methods of statis-tical analysis one employs for observations from the real line or Euclidean space. In fact, all the issues which one explores with linear data demand separate attention when one works with directional statistics. Accordingly, a separate theory has developed to answer the various ques- tions related to directional statistics. The relevant references are the books by Mardia (1972), Watson (1983), and Fisher et al. (1987). In addition to these books, the review papers by Mardia (1975a, 1975b, 1988), Rao (1984), and Jupp and Mardia (1989) provide surveys of what has been achieved in the study of directional statistics so far. See also latson (1939 ).This thesis explores some problems in directional statistics. The problems can broadly be classified into two categories : characterization and asymptotics.In what follows we mention briefly the contents of the thesis.Chernoff (1981) proved that if X~N(o,1), then for any absolutely continuous function g with E[g2(x)] finite


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