Brazilian Journal of Probability and Statistics

The gamma beta ratio distribution

Saralees Nadarajah

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The important problem of the ratio of gamma and beta distributed random variables is considered. Six motivating applications (from efficiency modeling, income modeling, clinical trials, hydrology, reliability and modeling of infectious diseases) are discussed. Exact expressions are derived for the probability density function, cumulative distribution function, hazard rate function, shape characteristics, moments, factorial moments, variance, skewness, kurtosis, conditional moments, L moments, characteristic function, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, percentiles, order statistics and the asymptotic distribution of the extreme values. Estimation procedures by the methods of moments and maximum likelihood are provided and their performances compared by simulation. For maximum likelihood estimation, the Fisher information matrix is derived and the case of censoring is considered. Finally, an application is discussed for efficiency of warning-time systems.

Article information

Source
Braz. J. Probab. Stat. Volume 26, Number 2 (2012), 178-207.

Dates
First available in Project Euclid: 23 January 2012

Permanent link to this document
http://projecteuclid.org/euclid.bjps/1327328084

Digital Object Identifier
doi:10.1214/10-BJPS128

Zentralblatt MATH identifier
1235.62017

Mathematical Reviews number (MathSciNet)
MR2880905

Citation

Nadarajah, Saralees. The gamma beta ratio distribution. Brazilian Journal of Probability and Statistics 26 (2012), no. 2, 178--207. doi:10.1214/10-BJPS128. http://projecteuclid.org/euclid.bjps/1327328084.


Export citation

References

  • Berry, D. A. (2004). Bayesian statistics and the ethics of clinical trials. Statistical Science 19, 175–187.
  • Berry, D. A. and Eick, S. G. (1995). Adaptive assignment versus balanced randomization in clinical trials: A decision analysis. Statistics in Medicine 14, 231–246.
  • Bonferroni, C. E. (1930). Elementi di Statistica Generale. Firenze: Seeber.
  • Clarke, R. T. (1979). Extension of annual streamflow record by correlation with precipitation subject to heterogeneous errors. Water Resources Research 15, 1081–1088.
  • Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. London: Chapman & Hall.
  • Exton, H. (1978). Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. New York: Halsted Press.
  • Fennelly, K. P., Davidow, A. L., Miller, S. L., Connell, N. and Ellner, J. J. (2004). Airborne infection with Bacillus anthracis—from mills to mail. Emerging Infectious Diseases 10, 996–1001.
  • Fennelly, K. P. and Nardell, E. A. (1998). The relative efficacy of respirators and room ventilation in preventing occupational tuberculosis. Infection Control and Hospital Epidemiology 19, 754–759.
  • Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products, 6th ed. San Diego: Academic Press.
  • Grandmont, J.-M. (1987). Distributions of preferences and the “law of demand”. Econometrica 55, 155–161.
  • Hoskings, J. R. M. (1990). L-moments: Analysis and estimation of distribution using linear combinations of order statistics. Journal of the Royal Statistical Society, Ser. B 52, 105–124.
  • Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions 1, 2nd ed. New York: Wiley.
  • Krishnaji, N. (1970). Characterization of the Pareto distribution through a model of underreported incomes. Econometrica 38, 251–255.
  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1987). Extremes and Related Properties of Random Sequences and Processes. New York: Springer.
  • Liao, C. M., Chang, C. F. and Liang, H. M. (2005). A probabilistic transmission dynamic model to assess indoor airborne infection risks. Risk Analysis 25, 1097–1107.
  • Milevsky, M. A. (1997). The present value of a stochastic perpetuity and the Gamma distribution. Insurance: Mathematics and Economics 20, 243–250.
  • Nadarajah, S. and Kotz, S. (2005). On the product and ratio of gamma and beta random variables. AStA Advances in Statistical Analysis 89, 435–449.
  • Nardell, E. A., Keegan, J., Cheney, S. A. and Etkind, S. C. (1991). Theoretical limits of protection achievable by building ventilation. American Review of Respiratory Disease 144, 302–306.
  • Nicas, M. (1996). Refining a risk model for occupational tuberculosis transmission. American Industrial Hygiene Association Journal 57, 16–22.
  • Nicas, M. (2000). Regulating the risk of tuberculosis transmission among health care workers. American Industrial Hygiene Association Journal 61, 334–339.
  • Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series, 13. Amsterdam: Gordon and Breach Science Publishers.
  • Rudnick, S. N. and Milton, D. K. (2003). Risk of indoor airborne infection transmission estimated from carbon dioxide concentration. Indoor Air 13, 237–245.
  • Sarabia, J. M., Castillo, E. and Slottje, D. J. (2002). Lorenz ordering between McDonald’s generalized functions of the income size distribution. Economics Letters 75, 265–270.
  • Silver, J., Slud, E. and Takamoto, K. (2002). Statistical equilibrium wealth distributions in an exchange economy with stochastic preferences. Journal of Economic Theory 106, 417–435.