Mixture of Two Lognormal Distributions (Alternative Parameterization)

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Description

Density, distribution function, quantile function, and random generation for a mixture of two lognormal distribution with parameters mean1, cv1, mean2, cv2, and p.mix.

Usage

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  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)

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.

mean1

vector of means of the first lognormal random variable. The default is
meanlog1=sqrt(exp(1) - 1).

cv1

vector of coefficient of variations of the first lognormal random variable. The default is sdlog1=sqrt(exp(1) - 1).

mean2

vector of means of the second lognormal random variable. The default is
mean2=sqrt(exp(1) - 1).

cv2

vector of coefficient of variations of the second lognormal random variable. The default is sdlog2=sqrt(exp(1) - 1).

p.mix

vector of probabilities between 0 and 1 indicating the mixing proportion. For rlnormMixAlt this must be a single, non-missing number.

Details

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.

Value

dlnormMixAlt gives the density, plnormMixAlt gives the distribution function, qlnormMixAlt gives the quantile function, and rlnormMixAlt generates random deviates.

Note

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).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

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.

See Also

LognormalAlt, LognormalMix, Lognormal, NormalMix, Probability Distributions and Random Numbers.

Examples

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  # 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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