# SEP1: The Skew Power exponential type 1-4 distribution for fitting... In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## 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 `mu.link`, with "identity" link as the default for the `mu` parameter. Other links are "inverse" 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 "log" link as the default for the `nu` parameter. Other links are "identity" and "inverse" `tau.link` Defines the `tau.link`, with "log" link as the default for the `tau` parameter. Other links are "inverse", 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

## 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 [email protected]

## 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.

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`, `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) ```

### Example output

```Loading required package: MASS

GAMLSS Family: SEP1 Skew exponential power (Azzalini type 1)
Link function for mu   : identity