# Log Normal distribution for fitting in GAMLSS

### Description

The functions `LOGNO`

and `LOGNO2`

define a `gamlss.family`

distribution to fits the log-Normal distribution.
The difference between them is that while `LOGNO`

retains the original parametrization for `mu`

, (identical to the normal distribution `NO`

) and therefore *mu=(-Inf,+Inf)*, the function `LOGNO2`

use `mu`

as the median, so *mu=(0,+Inf)*.

The function `LNO`

is more general and can fit a Box-Cox transformation
to data using the `gamlss()`

function.
In the `LOGNO`

(and `LOGNO2`

) there are two parameters involved `mu`

`sigma`

, while in the
`LNO`

there are three parameters `mu`

`sigma`

,
and the transformation parameter `nu`

.
The transformation parameter `nu`

in `LNO`

is a 'fixed' parameter (not estimated) and it has its default value equal to
zero allowing the fitting of the log-normal distribution as in `LOGNO`

.
See the example below on how to fix `nu`

to be a particular value.
In order to estimate (or model) the parameter `nu`

, use the `gamlss.family`

`BCCG`

distribution which uses a reparameterized version of the the Box-Cox transformation.
The functions `dLOGNO`

, `pLOGNO`

, `qLOGNO`

and `rLOGNO`

define the density, distribution function, quantile function and random
generation for the specific parameterization of the log-normal distribution.

The functions `dLOGNO2`

, `pLOGNO2`

, `qLOGNO2`

and `rLOGNO2`

define the density, distribution function, quantile function and random
generation when `mu`

is the median of the log-normal distribution.

The functions `dLNO`

, `pLNO`

, `qLNO`

and `rLNO`

define the density, distribution function, quantile function and random
generation for the specific parameterization of the log-normal distribution and more generally a Box-Cox transformation.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
LNO(mu.link = "identity", sigma.link = "log")
LOGNO(mu.link = "identity", sigma.link = "log")
LOGNO2(mu.link = "log", sigma.link = "log")
dLNO(x, mu = 1, sigma = 0.1, nu = 0, log = FALSE)
dLOGNO(x, mu = 0, sigma = 1, log = FALSE)
dLOGNO2(x, mu = 1, sigma = 1, log = FALSE)
pLNO(q, mu = 1, sigma = 0.1, nu = 0, lower.tail = TRUE, log.p = FALSE)
pLOGNO(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
pLOGNO2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLNO(p, mu = 1, sigma = 0.1, nu = 0, lower.tail = TRUE, log.p = FALSE)
qLOGNO(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLOGNO2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rLNO(n, mu = 1, sigma = 0.1, nu = 0)
rLOGNO(n, mu = 0, sigma = 1)
rLOGNO2(n, mu = 1, sigma = 1)
``` |

### Arguments

`mu.link` |
Defines the |

`sigma.link` |
Defines the |

`x,q` |
vector of quantiles |

`mu` |
vector of location parameter values |

`sigma` |
vector of scale parameter values |

`nu` |
vector of shape parameter values |

`log, log.p` |
logical; if TRUE, probabilities p are given as log(p). |

`lower.tail` |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x] |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

### Details

The probability density function in `LOGNO`

is defined as

*f(y|mu,sigma)=(1/(y*sqrt(2*pi)*sigma))*exp(-0.5*((log(y)-mu)/(sigma))^2)*

for *y>0*, *mu=(-Inf,+Inf)* and *σ>0*.

The probability density function in `LNO`

is defined as

*f(y|mu,sigma,nu)=(1/(sqrt(2*pi)*sigma))*(y^(nu-1))*exp(-((z-mu)/(2sigma))^2)*

where if *ν!=0* *z=(y^nu-1)/nu* else *z=\log(y)* and *z \sim N(0,σ^2)*,
for *y>0*, *μ>0*, *σ>0* and *ν=(-Inf,+Inf)*.

### Value

`LNO()`

returns a `gamlss.family`

object which can be used to fit a log-normal distribution in the `gamlss()`

function.
`dLNO()`

gives the density, `pLNO()`

gives the distribution
function, `qLNO()`

gives the quantile function, and `rLNO()`

generates random deviates.

### Warning

This is a two parameter fit for *mu* and *sigma* while *nu* is fixed.
If you wish to model *nu* use the gamlss family `BCCG`

.

### Note

*mu* is the mean of z (and also the median of y), the Box-Cox transformed variable and *sigma* is the standard deviation of z
and approximate the coefficient of variation of y

### Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

### References

Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations (with discussion), *J. R. Statist. Soc.* B., **26**, 211–252

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion),
*Appl. Statist.*, **54**, part 3, pp 507-554.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R.
*Journal of Statistical Software*, Vol. **23**, Issue 7, Dec 2007, http://www.jstatsoft.org/v23/i07.

### See Also

`gamlss.family`

, `BCCG`

### 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 | ```
LOGNO()# gives information about the default links for the log normal distribution
LOGNO2()
LNO()# gives information about the default links for the Box Cox distribution
# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGNO(x, mu=0), 0, 10)
curve(pLOGNO(x, mu=0), 0, 10)
curve(qLOGNO(x, mu=0), 0, 1)
Y<- rLOGNO(200)
hist(Y)
par(op)
# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGNO2(x, mu=1), 0, 10)
curve(pLOGNO2(x, mu=1), 0, 10)
curve(qLOGNO2(x, mu=1), 0, 1)
Y<- rLOGNO(200)
hist(Y)
par(op)
# library(gamlss)
# data(abdom)
# h1<-gamlss(y~cs(x), family=LOGNO, data=abdom)#fits the log-Normal distribution
# h2<-gamlss(y~cs(x), family=LNO, data=abdom) #should be identical to the one above
# to change to square root transformation, i.e. fix nu=0.5
# h3<-gamlss(y~cs(x), family=LNO, data=abdom, nu.fix=TRUE, nu.start=0.5)
``` |