The Bivariate Burr Distribution

Summary

The bivariate Burr distribution,

$${\text{F(x,y)}}\,{\text{ = }}\,{\text{1}}\, - \,{(1 + {{\text{x}}^{^{_{{{^{\text{b}}}_1}}}}})^{ - {\text{p}}}}\, - \,{(1 + {{\text{y}}^{^{_{^{\text{b}}2}}}})^{ - {\text{p}}}}\, + \,{(1 + {{\text{x}}^{^{_{{{^{\text{b}}}_1}}}}} + {{\text{y}}^{^{_{^{\text{b}}2}}}} + {\text{r}}{{\text{x}}^{^{_{{{^{\text{b}}}_1}}}}}{{\text{y}}^{^{_{^{\text{b}}2}}}})^{ - {\text{p}}}};\,{\text{x,}}\,{\text{y}}\,\underline \geqslant \,0,\,0\,\underline \leqslant \,{\text{r}}\,\underline \leqslant \,{\text{p}}\, + \,1;\,{\text{F(x,y)}}\,{\text{ = }}\,{\text{0}}\,{\text{elsewhere}}$$

is developed and investigated. Two special cases of the distribution occur when the parameter r = 0 and 1 respectively. For the limiting case r = 0, F(x,y) reduces to the bivariate case of the multivariate Burr distribution developed by Takahasi (1965). When r = 1, F(x,y) = F(x)·F(y), the independent case. The relationship of the bivariate Burr distribution and its marginals to the Pearson curves is discussed.