Lognormal3 {EnvStats}  R Documentation 
The ThreeParameter Lognormal Distribution
Description
Density, distribution function, quantile function, and random generation
for the threeparameter lognormal distribution with parameters meanlog
,
sdlog
, and threshold
.
Usage
dlnorm3(x, meanlog = 0, sdlog = 1, threshold = 0)
plnorm3(q, meanlog = 0, sdlog = 1, threshold = 0)
qlnorm3(p, meanlog = 0, sdlog = 1, threshold = 0)
rlnorm3(n, meanlog = 0, sdlog = 1, threshold = 0)
Arguments
x 
vector of quantiles. 
q 
vector of quantiles. 
p 
vector of probabilities between 0 and 1. 
n 
sample size. If 
meanlog 
vector of means of the distribution of the random variable on the log scale.
The default is 
sdlog 
vector of (positive) standard deviations of the random variable on the log scale.
The default is 
threshold 
vector of thresholds of the random variable on the log scale. The default
is 
Details
The threeparameter lognormal distribution is simply the usual twoparameter lognormal distribution with a location shift.
Let X
be a random variable with a threeparameter lognormal distribution
with parameters meanlog=
\mu
, sdlog=
\sigma
, and
threshold=
\gamma
. Then the random variable Y = X  \gamma
has a lognormal distribution with parameters
meanlog=
\mu
and sdlog=
\sigma
. Thus,

dlnorm3
callsdlnorm
using the argumentsx = x  threshold
,meanlog = meanlog
,sdlog = sdlog

plnorm3
callsplnorm
using the argumentsq = q  threshold
,meanlog = meanlog
,sdlog = sdlog

qlnorm3
callsqlnorm
using the argumentsq = q
,meanlog = meanlog
,sdlog = sdlog
and then adds the argumentthreshold
to the result. 
rlnorm3
callsrlnorm
using the argumentsn = n
,meanlog = meanlog
,sdlog = sdlog
and then adds the argumentthreshold
to the result.
The threshold parameter \gamma
affects only the location of the
threeparameter lognormal distribution; it has no effect on the variance
or the shape of the distribution.
Denote the mean, variance, and coefficient of variation of Y = X  \gamma
by:
E(Y) = \theta
Var(Y) = \eta^2
CV(Y) = \tau = \eta/\theta
Then the mean, variance, and coefficient of variation of X
are given by:
E(X) = \theta + \eta
Var(X) = \eta^2
CV(X) = \frac{\eta}{\theta + \gamma} = \frac{\tau \theta}{\theta + \gamma}
The relationships between the parameters \mu
, \sigma
,
\theta
, \eta
, and \tau
are as follows:
\theta = \beta \sqrt{\omega}
\eta = \beta \sqrt{\omega (\omega  1)}
\tau = \sqrt{\omega  1}
\mu = log(\frac{\theta}{\sqrt{\tau^2 + 1}})
\sigma = \sqrt{log(\tau^2 + 1)}
where
\beta = e^\mu, \omega = exp(\sigma^2)
Since quantiles of a distribution are preserved under monotonic transformations,
the median of X
is:
Median(X) = \gamma + \beta
Value
dlnorm3
gives the density, plnorm3
gives the distribution function,
qlnorm3
gives the quantile function, and rlnorm3
generates random
deviates.
Note
The twoparameter lognormal distribution is the distribution of a random variable whose logarithm is normally distributed. The two major characteristics of the twoparameter lognormal distribution are that it is bounded below at 0, and it is skewed to the right. The threeparameter lognormal distribution is a generalization of the twoparameter lognormal distribution in which the distribution is shifted so that the threshold parameter is some arbitrary number, not necessarily 0.
The threeparameter lognormal distribution was introduced by Wicksell (1917) in a study of the distribution of ages at first marriage. Both the two and threeparameter lognormal distributions have been used in a variety of fields, including economics and business, industry, biology, ecology, atmospheric science, and geology (Crow and Shimizu, 1988). Royston (1992) has discussed the application of the threeparameter lognormal distribution in the field of medicine.
The twoparameter 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 twoparameter lognormal distribution.
The threepararameter lognormal distribution starts to resemble a normal
distribution as the parameter \sigma
(the standard deviation of
log(X\gamma)
tends to 0.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
Aitchison, J., and J.A.C. Brown (1957). The Lognormal Distribution (with special references to its uses in economics). Cambridge University Press, London, 176pp.
Crow, E.L., and K. Shimizu. (1988). Lognormal Distributions: Theory and Applications. Marcel Dekker, New York, 387pp.
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.
Ott, W.R. (1990). A Physical Explanation of the Lognormality of Pollutant Concentrations. Journal of the Air and Waste Management Association 40, 1378–1383.
Ott, W.R. (1995). Environmental Statistics and Data Analysis. Lewis Publishers, Boca Raton, FL, Chapter 9.
Royston, J.P. (1992b). Estimation, Reference Ranges and Goodness of Fit for the ThreeParameter LogNormal Distribution. Statistics in Medicine 11, 897–912.
Wicksell, S.D. (1917). On Logarithmic Correlation with an Application to the Distribution of Ages at First Marriage. Medd. Lunds. Astr. Obs. 84, 1–21.
See Also
Lognormal, elnorm3
,
Probability Distributions and Random Numbers.
Examples
# Density of the threeparameter lognormal distribution with
# parameters meanlog=1, sdlog=2, and threshold=10, evaluated at 10.5:
dlnorm3(10.5, 1, 2, 10)
#[1] 0.278794
#
# The cdf of the threeparameter lognormal distribution with
# parameters meanlog=2, sdlog=3, and threshold=5, evaluated at 9:
plnorm3(9, 2, 3, 5)
#[1] 0.4189546
#
# The median of the threeparameter lognormal distribution with
# parameters meanlog=2, sdlog=3, and threshold=20:
qlnorm3(0.5, 2, 3, 20)
#[1] 27.38906
#
# Random sample of 3 observations from the threeparameter lognormal
# distribution with parameters meanlog=2, sdlog=1, and threshold=5.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(20)
rlnorm3(3, 2, 1, 5)
#[1] 18.6339749 0.8873173 39.0561521