The gamma beta ratio distribution. (English) Zbl 1235.62017

Summary: 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 extreme values. Estimation procedures by the methods of moments and maximum likelihood are provided and their performance is compared by simulations. For maximum likelihood estimation, the Fisher information matrix is derived and the case of censoring is considered. Finally, an application is discussed of the efficiency of warning-time systems.


62E15 Exact distribution theory in statistics
62E20 Asymptotic distribution theory in statistics
62F10 Point estimation
33C90 Applications of hypergeometric functions
62G32 Statistics of extreme values; tail inference
65C60 Computational problems in statistics (MSC2010)
Full Text: DOI Euclid


[1] Berry, D. A. (2004). Bayesian statistics and the ethics of clinical trials. Statistical Science 19 , 175-187. · Zbl 1057.62096
[2] 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. · Zbl 0972.62107
[3] Bonferroni, C. E. (1930). Elementi di Statistica Generale . Firenze: Seeber. · JFM 67.1057.03
[4] Clarke, R. T. (1979). Extension of annual streamflow record by correlation with precipitation subject to heterogeneous errors. Water Resources Research 15 , 1081-1088.
[5] Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics . London: Chapman & Hall. · Zbl 0334.62003
[6] Exton, H. (1978). Handbook of Hypergeometric Integrals: Theory, Applications, Tables, Computer Programs . New York: Halsted Press. · Zbl 0377.33001
[7] 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.
[8] 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.
[9] Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products , 6th ed. San Diego: Academic Press. · Zbl 0981.65001
[10] Grandmont, J.-M. (1987). Distributions of preferences and the “law of demand”. Econometrica 55 , 155-161. · Zbl 0606.90003
[11] 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. · Zbl 0703.62018
[12] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions 1 , 2nd ed. New York: Wiley. · Zbl 0811.62001
[13] Krishnaji, N. (1970). Characterization of the Pareto distribution through a model of underreported incomes. Econometrica 38 , 251-255.
[14] Leadbetter, M. R., Lindgren, G. and RootzĂ©n, H. (1987). Extremes and Related Properties of Random Sequences and Processes . New York: Springer. · Zbl 0518.60021
[15] 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. · Zbl 1216.62172
[16] Milevsky, M. A. (1997). The present value of a stochastic perpetuity and the Gamma distribution. Insurance: Mathematics and Economics 20 , 243-250. · Zbl 0903.60069
[17] 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. · Zbl 1379.62012
[18] 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.
[19] Nicas, M. (1996). Refining a risk model for occupational tuberculosis transmission. American Industrial Hygiene Association Journal 57 , 16-22.
[20] Nicas, M. (2000). Regulating the risk of tuberculosis transmission among health care workers. American Industrial Hygiene Association Journal 61 , 334-339.
[21] Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series , 1 - 3 . Amsterdam: Gordon and Breach Science Publishers. · Zbl 0733.00004
[22] Rudnick, S. N. and Milton, D. K. (2003). Risk of indoor airborne infection transmission estimated from carbon dioxide concentration. Indoor Air 13 , 237-245.
[23] 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. · Zbl 0994.60012
[24] 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. · Zbl 1033.91011
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