LognormalMixAlt.RdDensity, distribution function, quantile function, and random generation
for a mixture of two lognormal distribution with parameters
mean1, cv1, mean2, cv2, and p.mix.
dlnormMixAlt(x, mean1 = exp(1/2), cv1 = sqrt(exp(1) - 1),
mean2 = exp(1/2), cv2 = sqrt(exp(1) - 1), p.mix = 0.5)
plnormMixAlt(q, mean1 = exp(1/2), cv1 = sqrt(exp(1) - 1),
mean2 = exp(1/2), cv2 = sqrt(exp(1) - 1), p.mix = 0.5)
qlnormMixAlt(p, mean1 = exp(1/2), cv1 = sqrt(exp(1) - 1),
mean2 = exp(1/2), cv2 = sqrt(exp(1) - 1), p.mix = 0.5)
rlnormMixAlt(n, mean1 = exp(1/2), cv1 = sqrt(exp(1) - 1),
mean2 = exp(1/2), cv2 = sqrt(exp(1) - 1), p.mix = 0.5)vector of quantiles.
vector of quantiles.
vector of probabilities between 0 and 1.
sample size. If length(n) is larger than 1, then length(n)
random values are returned.
vector of means of the first lognormal random variable. The default is meanlog1=sqrt(exp(1) - 1).
vector of coefficient of variations of the first lognormal random variable.
The default is sdlog1=sqrt(exp(1) - 1).
vector of means of the second lognormal random variable. The default is mean2=sqrt(exp(1) - 1).
vector of coefficient of variations of the second lognormal random variable.
The default is sdlog2=sqrt(exp(1) - 1).
vector of probabilities between 0 and 1 indicating the mixing proportion.
For rlnormMixAlt this must be a single, non-missing number.
Let \(f(x; \eta, \theta)\) denote the density of a
lognormal random variable with parameters
mean=\(\eta\) and cv=\(\theta\). The density, \(g\), of a
lognormal mixture random variable with parameters mean1=\(\eta_1\),
cv1=\(\theta_1\), mean2=\(\eta_2\),
cv2=\(\theta_2\), and p.mix=\(p\) is given by:
$$g(x; \eta_1, \theta_1, \eta_2, \theta_2, p) =
(1 - p) f(x; \eta_1, \theta_1) + p f(x; \eta_2, \theta_2)$$
The default values for mean1 and cv1 correspond to a
lognormal distribution with parameters
meanlog=0 and sdlog=1. Similarly for the default values
of mean2 and cv2.
dlnormMixAlt gives the density, plnormMixAlt gives the distribution
function, qlnormMixAlt gives the quantile function, and
rlnormMixAlt generates random deviates.
Gilliom, R.J., and D.R. Helsel. (1986). Estimation of Distributional Parameters for Censored Trace Level Water Quality Data: 1. Estimation Techniques. Water Resources Research 22, 135-146.
Johnson, N. L., S. Kotz, and A.W. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, pp.53-54, and Chapter 8.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.
A lognormal mixture distribution is often used to model positive-valued data
that appear to be “contaminated”; that is, most of the values appear to
come from a single lognormal distribution, but a few “outliers” are
apparent. In this case, the value of mean2 would be larger than the
value of mean1, and the mixing proportion p.mix would be fairly
close to 0 (e.g., p.mix=0.1).
# Density of a lognormal mixture with parameters mean=2, cv1=3,
# mean2=4, cv2=5, p.mix=0.5, evaluated at 1.5:
dlnormMixAlt(1.5, mean1 = 2, cv1 = 3, mean2 = 4, cv2 = 5, p.mix = 0.5)
#> [1] 0.1436045
#[1] 0.1436045
#----------
# The cdf of a lognormal mixture with parameters mean=2, cv1=3,
# mean2=4, cv2=5, p.mix=0.5, evaluated at 1.5:
plnormMixAlt(1.5, mean1 = 2, cv1 = 3, mean2 = 4, cv2 = 5, p.mix = 0.5)
#> [1] 0.6778064
#[1] 0.6778064
#----------
# The median of a lognormal mixture with parameters mean=2, cv1=3,
# mean2=4, cv2=5, p.mix=0.5:
qlnormMixAlt(0.5, 2, 3, 4, 5, 0.5)
#> [1] 0.6978355
#[1] 0.6978355
#----------
# Random sample of 3 observations from a lognormal mixture with
# parameters mean1=2, cv1=3, mean2=4, cv2=5, p.mix=0.5.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(20)
rlnormMixAlt(3, 2, 3, 4, 5, 0.5)
#> [1] 0.70672151 14.43226313 0.05521329
#[1] 0.70672151 14.43226313 0.05521329