The Gamma Distribution (Alternative Parameterization)

Density, distribution function, quantile function, and random generation for the gamma distribution with parameters mean and cv.

dgammaAlt(x, mean, cv = 1, log = FALSE)
  pgammaAlt(q, mean, cv = 1, lower.tail = TRUE, log.p = FALSE)
  qgammaAlt(p, mean, cv = 1, lower.tail = TRUE, log.p = FALSE)
  rgammaAlt(n, mean, cv = 1)

Arguments

x: vector of quantiles.
q: vector of quantiles.
p: vector of probabilities between 0 and 1.
n: sample size. If length(n) is larger than 1, then length(n) random values are returned.
mean: vector of (positive) means of the distribution of the random variable.
cv: vector of (positive) coefficients of variation of the random variable.
log, log.p: logical; if TRUE, probabilities/densities $p$ are returned as $log(p)$.
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.

Details

Let $X$ be a random variable with a gamma distribution with parameters shape=$\alpha$ and scale=$\beta$. The relationship between these parameters and the mean (mean=$\mu$) and coefficient of variation (cv=$\tau$) of this distribution is given by: $$\alpha = \tau^{-2} \;\;\;\;\;\; (1)$$ $$\beta = \mu/\alpha \;\;\;\;\;\; (2)$$ $$\mu = \alpha\beta \;\;\;\;\;\; (3)$$ $$\tau = \alpha^{-1/2} \;\;\;\;\;\; (4)$$ Thus, the functions dgammaAlt, pgammaAlt, qgammaAlt, and rgammaAlt call the R functions dgamma, pgamma, qgamma, and rgamma, respectively, using the values for the shape and scale parameters given by: shape <- cv^-2, scale <- mean/shape.

Value

dgammaAlt gives the density, pgammaAlt gives the distribution function, qgammaAlt gives the quantile function, and rgammaAlt generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

References

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions, Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.

Singh, A., A.K. Singh, and R.J. Iaci. (2002). Estimation of the Exposure Point Concentration Term Using a Gamma Distribution. EPA/600/R-02/084. October 2002. Technology Support Center for Monitoring and Site Characterization, Office of Research and Development, Office of Solid Waste and Emergency Response, U.S. Environmental Protection Agency, Washington, D.C.

Singh, A., R. Maichle, and N. Armbya. (2010a). ProUCL Version 4.1.00 User Guide (Draft). EPA/600/R-07/041, May 2010. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.

Singh, A., N. Armbya, and A. Singh. (2010b). ProUCL Version 4.1.00 Technical Guide (Draft). EPA/600/R-07/041, May 2010. Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

The gamma distribution takes values on the positive real line. Special cases of the gamma are the exponential distribution and the chi-square distribution. Applications of the gamma include life testing, statistical ecology, queuing theory, inventory control and precipitation processes. A gamma distribution starts to resemble a normal distribution as the shape parameter $\alpha$ tends to infinity or the cv parameter $\tau$ tends to 0.

Some EPA guidance documents (e.g., Singh et al., 2002; Singh et al., 2010a,b) discourage using the assumption of a lognormal distribution for some types of environmental data and recommend instead assessing whether the data appear to fit a gamma distribution.

Examples

  # Density of a gamma distribution with parameters mean=10 and cv=2, 
  # evaluated at 7:

  dgammaAlt(7, mean = 10, cv = 2) 
#> [1] 0.02139335
  #[1] 0.02139335

  #----------

  # The cdf of a gamma distribution with parameters mean=10 and cv=2, 
  # evaluated at 12:

  pgammaAlt(12, mean = 10, cv = 2) 
#> [1] 0.7713307
  #[1] 0.7713307

  #----------

  # The 25'th percentile of a gamma distribution with parameters 
  # mean=10 and cv=2:

  qgammaAlt(0.25, mean = 10, cv = 2) 
#> [1] 0.1056871
  #[1] 0.1056871

  #----------

  # A random sample of 4 numbers from a gamma distribution with 
  # parameters mean=10 and cv=2. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(10) 
  rgammaAlt(4, mean = 10, cv = 2) 
#> [1] 3.772004230 1.889028078 0.002987823 8.179824976
  #[1] 3.772004230 1.889028078 0.002987823 8.179824976