Local dependence functions for some families of bivariate distributions and total positivity

The purpose of this paper is to investigate a very useful application of a certain local dependence function γf(x,y)γf(x,y), which was considered recently by Holland and Wang [20]. An interesting property of γf(x,y)γf(x,y) is that the underlying joint density f(x,y)f(x,y) is TP2TP2 (that is, totally positive of order 2) if and only if γf(x,y)≧0. This gives an elegant way to investigate the TP2TP2 property of any bivariate distribution. For the Saramanov family, the Ali–Mikhail–Haq family of bivariate distributions and the family of bivariate elliptical distributions, we derive the local dependence function and obtain conditions for f(x,y)f(x,y) to be TP2TP2. These families are quite rich and include many other large classes of bivariate distributions as their special cases. Similar conditions are obtained for bivariate distributions with exponential conditionals and bivariate distributions with Pareto conditionals.