Sign regularity of a generalized Cauchy kernel with applications

Abstract

This note provides a complete determination of the sign regularity properties of the 'Cauchy kernel' C-n(x,omega) = (ax+b omega+c)(n), which unifies and generalizes numerous results of sign regular-properties of various distribution families in the literature. As special case the total positivity of Pearson-type distributions including the family of Beta- and Pareto-densities is established. A further application is the construction of invariant UMP tests for interval hypotheses for the ratio of the variances of two normal distributions and the scaling parameters of two Gamma-or exponential-distributions. Finally, some more applications are given in the theory of optimal experimental designs.