A study on some geometrical and statistical properties of the continuous power-law distribution with a comparison of similar properties of a particular exponential distribution

Abstract

In literature, one can find diverse applications of the power-law distribution to model naturally occurring phenomena in the sciences. With the emerging field of complex networks this applicability has been observed and emphasised more and more. In the present paper several interesting as well as important properties of this distribution have been explored. First, it is shown that the totality of power-law distributions form a one-parameter family or a statistical model which turns out to be a Riemannian differentiable manifold. Closed form expressions are obtained for several information theoretically important measures such as differential entropy, information divergence and Fisher information which are interpreted to have geometrical and statistical meanings. Next, it is shown that the statistical manifold of power-law distributions forms an exponential family which is a very important aspect in mathematical statistics and information geometry including a number of other fields in the sciences. The parameter estimation problem is addressed using both maximum likelihood and entropy methods. The close relationship of the two in the sense of Statistics is elucidated. Finally, an example of an exponential distribution having the same information divergence and Fisher information as that of power-law distributions is given, thus having the same lower bound in the Cramer-Rao inequality. In this case, an approximate structural similarity can be expected between the two statistical manifolds.