Limit Distributions of Uncorrelated But Dependent Distributions on the Unit Square

Recently, bivariate distributions on the unit square [0,1]² generated by simple modifications of the uniform distribution on [0,l]² have been studied by the authors of this chapter (Johnson & Kotz, 1998; 1999). These distributions can be used as models for simulating distributions of cell counts in two-way contigency tables (Borkowf et. al., (1997). In our papers we were especially interested in measures of departure from uniformity in distributions with zero correlation between the variables but with some dependence between them. In our 1999 paper we studied a class of distributions that we called Square Tray distributions. These distributions are generated by just two levels of the probability density function (pdf), but can be extended to multiple-level square tray distributions in the way depicted (for three different levels) in Figure 9.1.