LognormalAlt: The Lognormal Distribution (Alternative Parameterization)

Description

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

Usage

dlnormAlt(x, mean = exp(1/2), cv = sqrt(exp(1) - 1), log = FALSE)
  plnormAlt(q, mean = exp(1/2), cv = sqrt(exp(1) - 1), 
      lower.tail = TRUE, log.p = FALSE)
  qlnormAlt(p, mean = exp(1/2), cv = sqrt(exp(1) - 1), 
      lower.tail = TRUE, log.p = FALSE)
  rlnormAlt(n, mean = exp(1/2), cv = sqrt(exp(1) - 1))

Value

dlnormAlt gives the density, plnormAlt gives the distribution function,

qlnormAlt gives the quantile function, and rlnormAlt generates random deviates.

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 (EnvStats@ProbStatInfo.com)

Details

Let $X$ be a random variable with a lognormal distribution with parameters meanlog=$\mu$ and sdlog=$\sigma$. That is, $\mu$ and $\sigma$ denote the mean and standard deviation of the random variable on the log scale. The relationship between these parameters and the mean (mean=$\theta$) and coefficient of variation (cv=$\tau$) of the distribution on the original scale is given by: $$\mu = log(\frac{\theta}{\sqrt{\tau^2 + 1}}) \;\;\;\; (1)$$ $$\sigma = [log(\tau^2 + 1)]^{1/2} \;\;\;\; (2)$$ $$\theta = exp[\mu + (\sigma^2/2)] \;\;\;\; (3)$$ $$\tau = [exp(\sigma^2) - 1]^{1/2} \;\;\;\; (4)$$ Thus, the functions dlnormAlt, plnormAlt, qlnormAlt, and rlnormAlt call the R functions dlnorm, plnorm, qlnorm, and rlnorm, respectively using the following values for the meanlog and sdlog parameters:
sdlog <- sqrt(log(1 + cv^2)),
meanlog <- log(mean) - (sdlog^2)/2

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.

Limpert, E., W.A. Stahel, and M. Abbt. (2001). Log-Normal Distributions Across the Sciences: Keys and Clues. BioScience 51, 341--352.

Ott, W.R. (1995). Environmental Statistics and Data Analysis. Lewis Publishers, Boca Raton, FL.

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.

Examples

Run this code

  # Density of the lognormal distribution with parameters 
  # mean=10 and cv=1, evaluated at 5: 

  dlnormAlt(5, mean = 10, cv = 1) 
  #[1] 0.08788173

  #----------

  # The cdf of the lognormal distribution with parameters mean=2 and cv=3, 
  # evaluated at 4: 

  plnormAlt(4, 2, 3) 
  #[1] 0.8879132

  #----------

  # The median of the lognormal distribution with parameters 
  # mean=10 and cv=1: 

  qlnormAlt(0.5, mean = 10, cv = 1) 
  #[1] 7.071068

  #----------

  # Random sample of 3 observations from a lognormal distribution with 
  # parameters mean=10 and cv=1. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(20) 
  rlnormAlt(3, mean = 10, cv = 1) 
  #[1] 18.615797  4.341402 31.265293

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