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

In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

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|n,\mu,\sigma\,\nu,\tau)==\frac{z}{\sigma} \Phi(\omega) \hspace{1mm} f_{EP}(z,0,1,\tau)

for -\infty < y < \infty , \mu=(-\infty,+\infty), \sigma>0, \nu=(-\infty,+\infty) and \tau>0. where z = \frac{y-\mu}{\sigma}, \omega = sign(z)|z|^{\tau/2}\nu \sqrt{2/\tau} and f_{EP}(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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9780429298547")}. 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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v023.i07")}.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) 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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

(see also https://www.gamlss.com/).

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))

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.