Statistics Seminars: The Cauchy-Schlomilch transformation, its extensions, and a useful analogue
14 April 2010 09:15 in Department of Mathematical Sciences, Durham University
Let g be the density of a symmetric univariate continuous distribution. I identify a wide class of "transformation of scale" functions t(x) such that functions of the form 2g(t(x)) are also densities; notice that I am not transforming the random variable associated with g and, importantly, that the normalising constant is just (that of g times) 2. Families of distributions are thereby generated by t and g. The basic version of this is the remarkable, simple but largely unknown, Cauchy-Schlomilch transformation. This turns out to be a special case of a more general approach to the problem. It is then seen that these "extended Cauchy- Schlomilch distributions" have close connections to a popular existing method of generating families of distributions and a number, perhaps a greater number, of interesting properties. Their "useful analogue" arises from application of much the same idea to distributions on the circle ... to much the same effect in a context where achieving the same effects as on the real line is not always as easy as it might seem!
An event within the Lecture Day with Prof. Balakrishnan, more info available at http://www.maths.dur.ac.uk/~dma0je/bala/