LognormalMix {EnvStats} | R Documentation |

## Mixture of Two Lognormal Distributions

### Description

Density, distribution function, quantile function, and random generation
for a mixture of two lognormal distribution with parameters
`meanlog1`

, `sdlog1`

, `meanlog2`

, `sdlog2`

, and `p.mix`

.

### Usage

```
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 |

`meanlog1` |
vector of means of the first lognormal random variable on the log scale.
The default is |

`sdlog1` |
vector of standard deviations of the first lognormal random variable on
the log scale. The default is |

`meanlog2` |
vector of means of the second lognormal random variable on the log scale.
The default is |

`sdlog2` |
vector of standard deviations of the second lognormal random variable on
the log scale. The default is |

`p.mix` |
vector of probabilities between 0 and 1 indicating the mixing proportion.
For |

### 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.

### 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`

).

### 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

Lognormal, NormalMix, Probability Distributions and Random Numbers.

### 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
#----------
# 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
#----------
# 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
#----------
# 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
```

*EnvStats*version 2.8.1 Index]