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.
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)Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "\(1/mu^2\)" and "log"
Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse" and "identity"
Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "\(1/nu^2\)" and "log"
Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "\(1/tau^2\)", and "identity
vector of quantiles
vector of location parameter values
vector of scale parameter values
vector of skewness nu parameter values
vector of kurtosis tau parameter values
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
vector of probabilities.
number of observations. If length(n) > 1, the length is
taken to be the number required
lower limit for the golden search to find quantiles from probabilities
upper limit for the golden search to find quantiles from probabilities
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.
The qSEP and rSEP are slow since they are relying on golden section for finding the quantiles
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.
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 http://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, 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.
# NOT RUN {
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))
# }
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