The Lognormal Distribution (Alternative Parameterization)

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

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))

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 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

Value

dlnormAlt gives the density, plnormAlt gives the distribution function, qlnormAlt gives the quantile function, and rlnormAlt generates random deviates.

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.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

The two-parameter lognormal distribution is the distribution of a random variable whose logarithm is normally distributed. The two major characteristics of the lognormal distribution are that it is bounded below at 0, and it is skewed to the right.

Because the empirical distribution of many variables is inherently positive and skewed to the right (e.g., size of organisms, amount of rainfall, size of income, etc.), the lognormal distribution has been widely applied in several fields, including economics, business, industry, biology, ecology, atmospheric science, and geology (Aitchison and Brown, 1957; Crow and Shimizu, 1988).

Gibrat (1930) derived the lognormal distribution from theoretical assumptions, calling it the "law of proportionate effect", but Kapteyn (1903) had described a machine that was the mechanical equivalent. The basic idea is that the Central Limit Theorem states that the distribution of the sum of several independent random variables tends to look like a normal distribution, no matter what the underlying distribution(s) of the original random variables, hence the product of several independent random variables tends to look like a lognormal distribution.

The lognormal distribution is often used to characterize chemical concentrations in the environment. Ott (1990) has shown mathematically how a series of successive random dilutions gives rise to a distribution that can be approximated by a lognormal distribution.

A lognormal distribution starts to resemble a normal distribution as the parameter $\sigma$ (the standard deviation of the log of the distribution) 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 the lognormal distribution with parameters 
  # mean=10 and cv=1, evaluated at 5: 

  dlnormAlt(5, mean = 10, cv = 1) 
#> [1] 0.08788173
  #[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
  #[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
  #[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
  #[1] 18.615797  4.341402 31.265293