# LOGISTIC: Logistic distribution for fitting a GAMLSS In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The function `LO()`, or equivalently `Logistic()`, defines the logistic distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function gamlss()

## Usage

 ```1 2 3 4 5``` ```LO(mu.link = "identity", sigma.link = "log") dLO(x, mu = 0, sigma = 1, log = FALSE) pLO(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE) qLO(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE) rLO(n, mu = 0, sigma = 1) ```

## Arguments

 `mu.link` Defines the `mu.link`, with "identity" link as the default for the mu parameter `sigma.link` Defines the `sigma.link`, with "log" link as the default for the sigma parameter `x,q` vector of quantiles `mu` vector of location parameter values `sigma` vector of scale 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 `length(n) > 1`, the length is taken to be the number required

## Details

Definition file for Logistic distribution.

f(y|mu,sigma)=(1/sigma)*exp((y-mu)/sigma)*((1+exp(-(y-mu)/sigma))^(-2))

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

## Value

`LO()` returns a `gamlss.family` object which can be used to fit a logistic distribution in the `gamlss()` function. `dLO()` gives the density, `pLO()` gives the distribution function, `qLO()` gives the quantile function, and `rLO()` generates random deviates for the logistic distribution. The latest functions are based on the equivalent `R` functions for logistic distribution.

## Note

mu is the mean and sigma*pi/sqrt(3) is the standard deviation for the logistic distribution

## Author(s)

Mikis Stasinopoulos [email protected], Bob Rigby and Calliope Akantziliotou

## References

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.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.

`gamlss.family`, `NO`, `TF`
 ```1 2 3 4 5 6 7 8``` ```LO()# gives information about the default links for the Logistic distribution plot(function(y) dLO(y, mu=10 ,sigma=2), 0, 20) plot(function(y) pLO(y, mu=10 ,sigma=2), 0, 20) plot(function(y) qLO(y, mu=10 ,sigma=2), 0, 1) # library(gamlss) # data(abdom) # h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=LO, data=abdom) # fits # plot(h) ```