Mixture of Two Lognormal Distributions

Density, distribution function, quantile function, and random generation for a mixture of two lognormal distribution with parameters meanlog1, sdlog1, meanlog2, sdlog2, and p.mix.

dlnormMix(x, meanlog1 = 0, sdlog1 = 1, meanlog2 = 0, sdlog2 = 1, p.mix = 0.5)
  plnormMix(q, meanlog1 = 0, sdlog1 = 1, meanlog2 = 0, sdlog2 = 1, p.mix = 0.5) 
  qlnormMix(p, meanlog1 = 0, sdlog1 = 1, meanlog2 = 0, sdlog2 = 1, p.mix = 0.5) 
  rlnormMix(n, meanlog1 = 0, sdlog1 = 1, meanlog2 = 0, sdlog2 = 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.
meanlog1: vector of means of the first lognormal random variable on the log scale. The default is meanlog1=0.
sdlog1: vector of standard deviations of the first lognormal random variable on the log scale. The default is sdlog1=1.
meanlog2: vector of means of the second lognormal random variable on the log scale. The default is meanlog2=0.
sdlog2: vector of standard deviations of the second lognormal random variable on the log scale. The default is sdlog2=1.
p.mix: vector of probabilities between 0 and 1 indicating the mixing proportion. For rlnormMix this must be a single, non-missing number.

Details

Let $f(x; \mu, \sigma)$ denote the density of a lognormal random variable with parameters meanlog=$\mu$ and sdlog=$\sigma$. The density, $g$, of a lognormal mixture random variable with parameters meanlog1=$\mu_1$, sdlog1=$\sigma_1$, meanlog2=$\mu_2$, sdlog2=$\sigma_2$, and p.mix=$p$ is given by: $$g(x; \mu_1, \sigma_1, \mu_2, \sigma_2, p) = (1 - p) f(x; \mu_1, \sigma_1) + p f(x; \mu_2, \sigma_2)$$

Value

dlnormMix gives the density, plnormMix gives the distribution function, qlnormMix gives the quantile function, and rlnormMix generates random deviates.

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.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

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 meanlog2 would be larger than the value of meanlog1, and the mixing proportion p.mix would be fairly close to 0 (e.g., p.mix=0.1). The value of the second standard deviation (sdlog2) may or may not be the same as the value for the first (sdlog1).

Examples

  # Density of a lognormal mixture with parameters meanlog1=0, sdlog1=1, 
  # meanlog2=2, sdlog2=3, p.mix=0.5, evaluated at 1.5: 

  dlnormMix(1.5, meanlog1 = 0, sdlog1 = 1, meanlog2 = 2, sdlog2 = 3, p.mix = 0.5) 
#> [1] 0.1609746
  #[1] 0.1609746

  #----------

  # The cdf of a lognormal mixture with parameters meanlog1=0, sdlog1=1, 
  # meanlog2=2, sdlog2=3, p.mix=0.2, evaluated at 4: 

  plnormMix(4, 0, 1, 2, 3, 0.2) 
#> [1] 0.8175281
  #[1] 0.8175281

  #----------

  # The median of a lognormal mixture with parameters meanlog1=0, sdlog1=1, 
  # meanlog2=2, sdlog2=3, p.mix=0.2: 

  qlnormMix(0.5, 0, 1, 2, 3, 0.2) 
#> [1] 1.156891
  #[1] 1.156891

  #----------

  # Random sample of 3 observations from a lognormal mixture with 
  # parameters meanlog1=0, sdlog1=1, meanlog2=3, sdlog2=4, p.mix=0.2. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(20) 
  rlnormMix(3, 0, 1, 2, 3, 0.2) 
#> [1] 0.08975283 1.07591103 7.85482514
  #[1] 0.08975283 1.07591103 7.85482514