# The Skew Power exponential type 1-4 distribution for fitting a GAMLSS

### Description

These functions define the Skew Power exponential type 1 to 4 distributions. All of them are four
parameter distributions and can be used to fit a GAMLSS model.
The functions `dSEP1`

, `dSEP2`

, `dSEP3`

and `dSEP4`

define the probability distribution functions,
the functions `pSEP1`

, `pSEP2`

, `pSEP3`

and `pSEP4`

define the cumulative distribution functions
the functions `qSEP1`

, `qSEP2`

, `qSEP3`

and `qSEP4`

define the inverse cumulative distribution functions and
the functions `rSEP1`

, `rSEP2`

, `rSEP3`

and `rSEP4`

define the random generation for the Skew exponential power
distributions.

### Usage

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 29 30 31 32 33 34 | ```
SEP1(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
dSEP1(x, mu = 0, sigma = 1, nu = 0, tau = 2, log = FALSE)
pSEP1(q, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE,
log.p = FALSE)
qSEP1(p, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE,
log.p = FALSE)
rSEP1(n, mu = 0, sigma = 1, nu = 0, tau = 2)
SEP2(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
dSEP2(x, mu = 0, sigma = 1, nu = 0, tau = 2, log = FALSE)
pSEP2(q, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE,
log.p = FALSE)
qSEP2(p, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE,
log.p = FALSE)
rSEP2(n, mu = 0, sigma = 1, nu = 0, tau = 2)
SEP3(mu.link = "identity", sigma.link = "log", nu.link = "log",
tau.link = "log")
dSEP3(x, mu = 0, sigma = 1, nu = 2, tau = 2, log = FALSE)
pSEP3(q, mu = 0, sigma = 1, nu = 2, tau = 2, lower.tail = TRUE,
log.p = FALSE)
qSEP3(p, mu = 0, sigma = 1, nu = 2, tau = 2, lower.tail = TRUE,
log.p = FALSE)
SEP4(mu.link = "identity", sigma.link = "log", nu.link = "log",
tau.link = "log")
dSEP4(x, mu = 0, sigma = 1, nu = 2, tau = 2, log = FALSE)
pSEP4(q, mu = 0, sigma = 1, nu = 2, tau = 2, lower.tail = TRUE,
log.p = FALSE)
qSEP4(p, mu = 0, sigma = 1, nu = 2, tau = 2, lower.tail = TRUE,
log.p = FALSE)
rSEP4(n, mu = 0, sigma = 1, nu = 2, tau = 2)
``` |

### Arguments

`mu.link` |
Defines the |

`sigma.link` |
Defines the |

`nu.link` |
Defines the |

`tau.link` |
Defines the |

`x,q` |
vector of quantiles |

`mu` |
vector of location parameter values |

`sigma` |
vector of scale parameter values |

`nu` |
vector of skewness |

`tau` |
vector of kurtosis |

`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 of the Skew Power exponential distribution type 2, (`SEP2`

), is defined as

*f(y|mu,sigma,nu,tau)=nu/(sigma*(1+nu^2)*2^(1/tau) Gamma(1+1/tau))*(exp(-.5*abs(nu(y-mu)/sigma))^tau*I(y<mu)+exp(-.5*abs((y-mu)/sigma*nu))^tau*I(y>=mu))*

for *0<y<0*,
*mu=(-Inf,+Inf)*,
*sigma>0*,
*nu>0)* and
*tau>0*.

### Value

`SEP2()`

returns a `gamlss.family`

object which can be used to fit the SEP2 distribution in the `gamlss()`

function.
`dSEP2()`

gives the density, `pSEP2()`

gives the distribution
function, `qSEP2()`

gives the quantile function, and `rSEP2()`

generates random deviates.

### Author(s)

Bob Rigby and Mikis Stasinopoulos mikis.stasinopoulos@gamlss.org

### References

Fernadez C., Osiewalski J. and Steel M.F.J.(1995) Modelling and inference with v-spherical distributions.
*JASA*, **90**, pp 1331-1340.

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`

, `SEP`

### Examples

1 2 3 4 5 6 7 8 9 10 | ```
SEP1()
curve(dSEP4(x, mu=5 ,sigma=1, nu=2, tau=1.5), -2, 10,
main = "The SEP4 density mu=5 ,sigma=1, nu=1, tau=1.5")
# library(gamlss)
#y<- rSEP4(100, mu=5, sigma=1, nu=2, tau=1.5);hist(y)
#m1<-gamlss(y~1, family=SEP1, n.cyc=50)
#m2<-gamlss(y~1, family=SEP2, n.cyc=50)
#m3<-gamlss(y~1, family=SEP3, n.cyc=50)
#m4<-gamlss(y~1, family=SEP4, n.cyc=50)
#GAIC(m1,m2,m3,m4)
``` |