Source: Braz. J. Probab. Stat.
Volume 26, Number 2
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.
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.
Mathematical Reviews (MathSciNet): MR370837
Exton, H. (1978). Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs. New York: Halsted Press.
Mathematical Reviews (MathSciNet): MR474684
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.
Mathematical Reviews (MathSciNet): MR875521
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.
Mathematical Reviews (MathSciNet): MR691492
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, 1–3. Amsterdam: Gordon and Breach Science Publishers.
Mathematical Reviews (MathSciNet): MR888165
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.