Mixture of Two Lognormal Distributions (Alternative Parameterization)
Density, 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)x |
vector of quantiles. |
q |
vector of quantiles. |
p |
vector of probabilities between 0 and 1. |
n |
sample size. If |
mean1 |
vector of means of the first lognormal random variable. The default is |
cv1 |
vector of coefficient of variations of the first lognormal random variable.
The default is |
mean2 |
vector of means of the second lognormal random variable. The default is |
cv2 |
vector of coefficient of variations of the second lognormal random variable.
The default is |
p.mix |
vector of probabilities between 0 and 1 indicating the mixing proportion.
For |
Let f(x; η, θ) denote the density of a
lognormal random variable with parameters
mean=η and cv=θ. The density, g, of a
lognormal mixture random variable with parameters mean1=η_1,
cv1=θ_1, mean2=η_2,
cv2=θ_2, and p.mix=p is given by:
g(x; η_1, θ_1, η_2, θ_2, p) = (1 - p) f(x; η_1, θ_1) + p f(x; η_2, θ_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.
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).
Steven P. Millard (EnvStats@ProbStatInfo.com)
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.
# 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 #---------- # 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 #---------- # 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 #---------- # 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
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