R: The Skew Power exponential type 1-4 distribution for fitting...

SEP1 {gamlss.dist} | R Documentation

## 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

    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

### 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.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

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, https://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.

### See Also

`gamlss.family`, `SEP`

### Examples

    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)

[Package gamlss.dist version 5.3-2 Index]