EnvStats (version 2.7.0)

GammaAlt: The Gamma Distribution (Alternative Parameterization)

Description

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

Usage

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)

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.

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, \(P[X > x]\).

Author

Steven P. Millard ()

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.

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.

See Also

GammaDist, egammaAlt, Probability Distributions and Random Numbers.

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

  #----------

  # 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

  #----------

  # 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

  #----------

  # 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