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3 results

  • A characterization of the uniform distribution on the circle in the analysis of directional data

  • M. S. Bingham
  • Journal:
    Journal of Applied Probability / Volume 15 / Issue 4 / December 1978
    Published online by Cambridge University Press:
    14 July 2016, pp. 852-857
    Print publication:
    December 1978

  • It was conjectured some time ago by K. V. Mardia that if, for random samples of some fixed size N ≧ 2 from a given non-degenerate circular population, the sample mean direction and the sample resultant length are independently distributed, then the population must be uniformly distributed round the circle. In this paper it is shown that, apart from one minor exception, Mardia's conjecture is true in the case N = 2. No regularity conditions are necessary for the proof. The same problem has been studied, subject to regularity conditions, by Kent, Mardia and Rao (1976) for all sample sizes N ≧ 2 except N = 4.

  • Distances on circles, toruses and spheres

  • B. W. Silverman
  • Journal:
    Journal of Applied Probability / Volume 15 / Issue 1 / March 1978
    Published online by Cambridge University Press:
    14 July 2016, pp. 136-143
    Print publication:
    March 1978

  • Families of heavily dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj) for some suitable function g of two arguments and independent uniformly distributed random variables Y1, Y2, ··· on the circle, the torus or the sphere. The weak convergence of the empirical distribution process is discussed. The particular case of distances between pairs of observations on the circle is considered in greater detail.