The Bivariate Burr Distribution

• Frederick C. Durling
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 17)

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.

Keywords

Burr distribution general system of distributions Pearson system of frequency curves bivariate Burr distribution

References

1. 
Bryson, M. C. (1974). Technometrics 16, 61–68.
2. 
Burr, I. W. (1942). Ann. of Math. Statist. 13, 215–232.
3. 
Burr, I. W. (1967a). Industrial Quality Control 23, 563–569.Google Scholar
4. 
Burr, I. W. (1967b). Technometrics 9, 647–651.
5. 
Burr, I. W. (1973). Comm. Statist. 2., 1–21.Google Scholar
6. 
Burr, I. W. and Cisiak, P. J. (1968). J. Amer. Statist. Assoc. 63, 627–643.
7. 
Craig, C. C. (1936). Ann. Math. Statist. 7, 16–28.
8. 
Durling, F. C. (1969). Bivariate Probit, Logit and Burrit Analysis. Themis Signal Analysis Statistics Research Pro-gram, Tech. Rept. 41Google Scholar
9. 
Durling, F. C., Owen, D. B., and Drane, J. W. (1970). Ann. Math. Statist. 41, 1135.Google Scholar
10. 
Gradshteyn, I. S. and Ryshik, I. M. (1965). Table of Integrals, Series and Products. Academic Press, New York and London.Google Scholar
11. 
Hatke, Sister M. A. (1949). Ann. Math. Statist. 20, 461–463.
12. 
Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. John Wiley and Sons, Inc., New York and London.
13. 
Pearson, E. S. and Hartley, H. O. (1966). Biometrika Tables for Statisticians, Vol. I, 3rd ed. Cambridge University Press, Cambridge.
14. 
Seibert, G. B., Jr. (1970). Estimation and Confidence Intervals for Quantal Response or Sensitivity Data. Themis Signal Analysis Statistics Research Program, Tech. Rept. 66, Department of Statistics, Southern Methodist University, Dallas.Google Scholar
15. Takahasi, K . (1965). Ann. Inst. Statist. Math. (Tokyo) 17, 257–260.
16. 
Wheeler, D. J. (1970). An Alternative to an F-Test on Variances. Themis Signal Analysis Statistics Research Program, Tech. Rept. 76, Department of Statistics, Southern Methodist University, DallasGoogle Scholar

© D. Reidel Publishing Company, Dordrecht-Holland 1975

Authors and Affiliations

• Frederick C. Durling
• 1
1. 1.University of WaikatoHamiltonNew Zealand

Personalised recommendations

Citepaper 