# SEP: The Skew Power exponential (SEP) distribution for fitting a... In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

This function defines the Skew Power exponential (SEP) distribution, a four parameter distribution, for a `gamlss.family` object to be used for a GAMLSS fitting using the function `gamlss()`. The functions `dSEP`, `pSEP`, `qSEP` and `rSEP` define the density, distribution function, quantile function and random generation for the Skew Power exponential (SEP) distribution.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```SEP(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link = "log") dSEP(x, mu = 0, sigma = 1, nu = 0, tau = 2, log = FALSE) pSEP(q, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, log.p = FALSE) qSEP(p, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, log.p = FALSE, lower.limit = mu - 5 * sigma, upper.limit = mu + 5 * sigma) rSEP(n, mu = 0, sigma = 1, nu = 0, tau = 2) ```

## Arguments

 `mu.link` Defines the `mu.link`, with "identity" link as the default for the `mu` parameter. Other links are "1/mu^2" and "log" `sigma.link` Defines the `sigma.link`, with "log" link as the default for the `sigma` parameter. Other links are "inverse" and "identity" `nu.link` Defines the `nu.link`, with "identity" link as the default for the `nu` parameter. Other links are "1/nu^2" and "log" `tau.link` Defines the `tau.link`, with "log" link as the default for the `tau` parameter. Other links are "1/tau^2", and "identity `x,q` vector of quantiles `mu` vector of location parameter values `sigma` vector of scale parameter values `nu` vector of skewness `nu` parameter values `tau` vector of kurtosis `tau` 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 `lower.limit` lower limit for the golden search to find quantiles from probabilities `upper.limit` upper limit for the golden search to find quantiles from probabilities

## Details

The probability density function of the Skew Power exponential distribution, (`SEP`), is defined as

f(y|mu,sigma,nu,tau)=(z/sigma)*pnorm(w)*dPE(z,0,1,tau)

for 0<y<0, mu=(-Inf,+Inf), sigma>0, nu=(-Inf,+Inf) and tau>0. where z=(y-mu)/(sigma), w=sign(z)|z|^(t/2) *nu*sqrt(2/tau) and dPE(z,0,1,tau) is the pdf of an Exponential Power distribution.

## Value

`SEP()` returns a `gamlss.family` object which can be used to fit the SEP distribution in the `gamlss()` function. `dSEP()` gives the density, `pSEP()` gives the distribution function, `qSEP()` gives the quantile function, and `rSEP()` generates random deviates.

## Warning

The qSEP and rSEP are slow since they are relying on golden section for finding the quantiles

## Author(s)

Bob Rigby and Mikis Stasinopoulos [email protected]

## References

Diciccio, T. J. and Mondi A. C. (2004). Inferential Aspects of the Skew Exponential Power distribution., JASA, 99, 439-450.

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`, `JSU`, `BCT`

## Examples

 ```1 2 3 4 5 6 7 8``` ```SEP() # plot(function(x)dSEP(x, mu=0,sigma=1, nu=1, tau=2), -5, 5, main = "The SEP density mu=0,sigma=1,nu=1, tau=2") plot(function(x) pSEP(x, mu=0,sigma=1,nu=1, tau=2), -5, 5, main = "The BCPE cdf mu=0, sigma=1, nu=1, tau=2") dat <- rSEP(100,mu=10,sigma=1,nu=-1,tau=1.5) # library(gamlss) # gamlss(dat~1,family=SEP, control=gamlss.control(n.cyc=30)) ```

### Example output

```Loading required package: MASS

GAMLSS Family: SEP Skew Exponential Power
Link function for mu   : identity