R: The Skew Power exponential (SEP) distribution for fitting a...

SEP {gamlss.dist} | R Documentation

## The Skew Power exponential (SEP) distribution for fitting a GAMLSS

### 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

    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

### 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.

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`, `JSU`, `BCT`

### Examples

    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))

[Package gamlss.dist version 5.3-2 Index]