# SN2: Skew Normal Type 2 distribution for fitting a GAMLSS In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The function `SN2()` defines the Skew Normal Type 2 distribution, a three parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`, with parameters `mu`, `sigma` and `nu`. The functions `dSN2`, `pSN2`, `qSN2` and `rSN2` define the density, distribution function, quantile function and random generation for the `SN2` parameterization of the Skew Normal Type 2 distribution.

## Usage

 ```1 2 3 4 5``` ```SN2(mu.link = "identity", sigma.link = "log", nu.link = "log") dSN2(x, mu = 0, sigma = 1, nu = 2, log = FALSE) pSN2(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE) qSN2(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE) rSN2(n, mu = 0, sigma = 1, nu = 2) ```

## Arguments

 `mu.link` Defines the `mu.link`, with "‘identity"’ links the default for the mu parameter `sigma.link` Defines the `sigma.link`, with "‘log"’ as the default for the sigma parameter `nu.link` Defines the `nu.link`, with "‘log"’ as the default for the sigma parameter `x, q` vector of quantiles `mu` vector of location parameter values `sigma` vector of scale parameter values `nu` 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

The parameterization of the Skew Normal Type 2 distribution in the function `SN2` is ...

## Value

returns a gamlss.family object which can be used to fit a Skew Normal Type 2 distribution in the `gamlss()` function.

## Note

This is a special case of the Skew Exponential Power type 3 distribution (`SEP3`)where `tau=2`.

## Author(s)

Mikis Stasinopoulos, Bob Rigby and Fiona McElduff.

## 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`
 ```1 2 3 4 5 6 7 8 9``` ```par(mfrow=c(2,2)) y<-seq(-3,3,0.2) plot(y, dSN2(y), type="l" , lwd=2) q<-seq(-3,3,0.2) plot(q, pSN2(q), ylim=c(0,1), type="l", lwd=2) p<-seq(0.0001,0.999,0.05) plot(p, qSN2(p), type="l", lwd=2) dat <- rSN2(100) hist(rSN2(100), nclass=30) ```