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.
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)
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.
Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse" 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 "log" link as the default for the nu parameter. Other links are "identity" and "inverse"
Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "inverse", 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
Bob Rigby and Mikis Stasinopoulos
The probability density function of the Skew Power exponential distribution type 1, (SEP1), is defined as
$$f_Y(y|\mu,\sigma\,\nu,\tau)=\frac{2}{\sigma} f_{Z_1}(z) F_{Z_1}(\nu z)$$
for \( -\infty < y < \infty \), \(-\infty<\mu <\infty\), \(\sigma>0\), \(-\infty<\nu <\infty\), and \(\tau>0\) where \(z=(y-\mu)/\sigma\), and \(Z_1 \sim \texttt{PE2}(0, \tau^{1/\tau }, \tau) \), see pp 401-402 of Rigby et al. (2019).
The probability density function of the Skew Power exponential distribution type 2, (SEP2), is defined as
$$f_Y(y|\mu,\sigma\,\nu,\tau)=\frac{2}{\sigma} f_{Z_1}(z) \Phi_{Z_1}(\omega)$$
for \( -\infty < y < \infty \), \(-\infty<\mu <\infty\), \(\sigma>0\), \(-\infty<\nu <\infty\), and \(\tau>0\) where \(z=(y-\mu)/\sigma\), and \(\omega= \texttt{sign}(z) |z|^{\tau/2} \nu \sqrt{2/\tau}\), and \(Z_1 \sim \texttt{PE2}(0, \tau^{1/\tau }, \tau) \), see pp 402-404 of Rigby et al. (2019).
For SEP3 and SEP3 see pp 404-406 and pp 407-408 of Rigby et al. (2019), respectively.
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, 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, tools:::Rd_expr_doi("10.18637/jss.v023.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. tools:::Rd_expr_doi("10.1201/b21973")
(see also https://www.gamlss.com/).
gamlss.family, SEP
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)
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