Lognormal3 {EnvStats} | R Documentation |
Density, distribution function, quantile function, and random generation
for the three-parameter lognormal distribution with parameters meanlog
,
sdlog
, and threshold
.
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)
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 |
The three-parameter lognormal distribution is simply the usual two-parameter lognormal distribution with a location shift.
Let X
be a random variable with a three-parameter 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
calls dlnorm
using the arguments
x = x - threshold
, meanlog = meanlog
,
sdlog = sdlog
plnorm3
calls plnorm
using the arguments
q = q - threshold
, meanlog = meanlog
, sdlog = sdlog
qlnorm3
calls qlnorm
using the arguments
q = q
, meanlog = meanlog
, sdlog = sdlog
and then adds
the argument threshold
to the result.
rlnorm3
calls rlnorm
using the arguments
n = n
, meanlog = meanlog
, sdlog = sdlog
and then adds
the argument threshold
to the result.
The threshold parameter \gamma
affects only the location of the
three-parameter 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
dlnorm3
gives the density, plnorm3
gives the distribution function,
qlnorm3
gives the quantile function, and rlnorm3
generates random
deviates.
The two-parameter lognormal distribution is the distribution of a random variable whose logarithm is normally distributed. The two major characteristics of the two-parameter lognormal distribution are that it is bounded below at 0, and it is skewed to the right. The three-parameter lognormal distribution is a generalization of the two-parameter lognormal distribution in which the distribution is shifted so that the threshold parameter is some arbitrary number, not necessarily 0.
The three-parameter lognormal distribution was introduced by Wicksell (1917) in a study of the distribution of ages at first marriage. Both the two- and three-parameter 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 three-parameter lognormal distribution in the field of medicine.
The two-parameter 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 two-parameter lognormal distribution.
The three-pararameter lognormal distribution starts to resemble a normal
distribution as the parameter \sigma
(the standard deviation of
log(X-\gamma)
tends to 0.
Steven P. Millard (EnvStats@ProbStatInfo.com)
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 Three-Parameter Log-Normal 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.
Lognormal, elnorm3
,
Probability Distributions and Random Numbers.
# Density of the three-parameter 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 three-parameter 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 three-parameter 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 three-parameter 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