LognormalMixAlt: Mixture of Two Lognormal Distributions (Alternative Parameterization)

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

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)

Value

dlnormMixAlt gives the density, plnormMixAlt gives the distribution function, qlnormMixAlt gives the quantile function, and

rlnormMixAlt generates random deviates.

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.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

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.

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.

Examples

Run this code

  # 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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