## Brazilian Journal of Probability and Statistics

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

### The gamma beta ratio distribution

#### 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**

https://projecteuclid.org/euclid.bjps/1327328084

**Digital Object Identifier**

doi:10.1214/10-BJPS128

**Mathematical Reviews number (MathSciNet)**

MR2880905

**Zentralblatt MATH identifier**

1235.62017

**Keywords**

Beta distribution estimation gamma distribution moments ratio of random variables

#### Citation

Nadarajah, Saralees. The gamma beta ratio distribution. Braz. J. Probab. Stat. 26 (2012), no. 2, 178--207. doi:10.1214/10-BJPS128. https://projecteuclid.org/euclid.bjps/1327328084.