The Truncated Lognormal Distribution

Density, distribution function, quantile function, and random generation for the truncated lognormal distribution with parameters meanlog, sdlog, min, and max.

dlnormTrunc(x, meanlog = 0, sdlog = 1, min = 0, max = Inf)
  plnormTrunc(q, meanlog = 0, sdlog = 1, min = 0, max = Inf)
  qlnormTrunc(p, meanlog = 0, sdlog = 1, min = 0, max = Inf)
  rlnormTrunc(n, meanlog = 0, sdlog = 1, min = 0, max = Inf)

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.
meanlog: vector of means of the distribution of the non-truncated random variable on the log scale. The default is meanlog=0.
sdlog: vector of (positive) standard deviations of the non-truncated random variable on the log scale. The default is sdlog=1.
min: vector of minimum values for truncation on the left. The default value is min=0.
max: vector of maximum values for truncation on the right. The default value is max=Inf.

Details

See the help file for the lognormal distribution for information about the density and cdf of a lognormal distribution.

Probability Density and Cumulative Distribution Function
Let $X$ denote a random variable with density function $f(x)$ and cumulative distribution function $F(x)$, and let $Y$ denote the truncated version of $X$ where $Y$ is truncated below at min=$A$ and above atmax=$B$. Then the density function of $Y$, denoted $g(y)$, is given by: $$g(y) = frac{f(y)}{F(B) - F(A)}, A \le y \le B$$ and the cdf of Y, denoted $G(y)$, is given by:

$G(y) =$	0	for $y < A$
	$\frac{F(y) - F(A)}{F(B) - F(A)}$	for $A \le y \le B$
	1	for $y > B$

Quantiles
The $p^{th}$ quantile $y_p$ of $Y$ is given by:

$y_p =$	$A$	for $p = 0$
	$F^{-1}\{p[F(B) - F(A)] + F(A)\} $	for $0 < p < 1$
	$B$	for $p = 1$

Random Numbers
Random numbers are generated using the inverse transformation method: $$y = G^{-1}(u)$$ where $u$ is a random deviate from a uniform $[0, 1]$ distribution.

Value

dlnormTrunc gives the density, plnormTrunc gives the distribution function, qlnormTrunc gives the quantile function, and rlnormTrunc 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.

Schneider, H. (1986). Truncated and Censored Samples from Normal Populations. Marcel Dekker, New York, Chapter 2.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

A truncated lognormal distribution is sometimes used as an input distribution for probabilistic risk assessment.

Examples

  # Density of a truncated lognormal distribution with parameters 
  # meanlog=1, sdlog=0.75, min=0, max=10, evaluated at 2 and 4:

  dlnormTrunc(c(2, 4), 1, 0.75, 0, 10) 
#> [1] 0.2551219 0.1214676
  #[1] 0.2551219 0.1214676

  #----------

  # The cdf of a truncated lognormal distribution with parameters 
  # meanlog=1, sdlog=0.75, min=0, max=10, evaluated at 2 and 4:

  plnormTrunc(c(2, 4), 1, 0.75, 0, 10) 
#> [1] 0.3558867 0.7266934
  #[1] 0.3558867 0.7266934

  #----------

  # The median of a truncated lognormal distribution with parameters 
  # meanlog=1, sdlog=0.75, min=0, max=10:

  qlnormTrunc(.5, 1, 0.75, 0, 10) 
#> [1] 2.614945
  #[1] 2.614945

  #----------

  # A random sample of 3 observations from a truncated lognormal distribution 
  # with parameters meanlog=1, sdlog=0.75, min=0, max=10. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(20) 
  rlnormTrunc(3, 1, 0.75, 0, 10) 
#> [1] 5.754805 4.372218 1.706815
  #[1] 5.754805 4.372218 1.706815

\(G(y) =\)	0	for \(y < A\)
	\(\frac{F(y) - F(A)}{F(B) - F(A)}\)	for \(A \le y \le B\)
	1	for \(y > B\)

\(y_p =\)	\(A\)	for \(p = 0\)
	\(F^{-1}\{p[F(B) - F(A)] + F(A)\} \)	for \(0 < p < 1\)
	\(B\)	for \(p = 1\)