Colloquia — Summer 2001
Monday, July 2, 2001
Some New Distributions in Statistics
University of California, Santa Barbara
A. N. V. Rao
Speaker is a candidate for Asst. Prof. in Statistics.
I shall talk about two of my most recent research products:
- the Planck distribution (joint work with Professor Leipnik, Department of Mathematics, University of California, Santa Barbara);
- some new elliptical distributions (joint work with Professor Kotz, Department of Engineering Management and Systems Engineering, George Washington University, Washington, D.C.).
Planck is a well-known physicist. He discovered a distribution that generalizes the well-known Gamma distribution. This distribution, which we refer to as the Planck distribution, has been around in the physics literature for many years. However, to the best of our knowledge, it does not appear to have been noted in the statistics literature. In the first part of my talk, I shall introduce the Planck distribution and derive its basic properties such as the characteristic function, moments, likelihood, maximum likelihood equations and the Fisher information matrix. Calculations involve the “Lerch” function.
The last twenty years have seen a vigorous development of multivariate elliptical distributions as direct generalizations of the multivariate normal distribution, which has dominated statistical theory and applications for almost a century. One of the most attractive elliptical distributions known in the statistics literature is the so-called Kotz type distribution (named after my co-author). It provides a most natural generalization of the multivariate normal distribution. In the second part of my talk, I shall note that the Kotz type distribution can be interpreted as being generated by the Type III extreme value distribution. Based on this observation, I shall develop two new elliptical distributions based on the Type I and Type II extreme value distributions and show results on their structural properties.