# LognormalMixAlt: Mixture of Two Lognormal Distributions (Alternative... In EnvStats: Package for Environmental Statistics, Including US EPA Guidance

## 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

 ```1 2 3 4 5 6 7 8``` ``` 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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31``` ``` # 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) # 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) # 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) # 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) # 0.70672151 14.43226313 0.05521329 ```

EnvStats documentation built on Oct. 23, 2020, 6:41 p.m.