The Skew t type 3 distribution Jones and Faddy (2003).
The functions dST3, pST3, qST3 and rST3 define the density, distribution function,
quantile function and random generation for the skew t distribution type 3.
The SST is a reparametrised version of dST3 where sigma is the standard deviation of the distribution.
ST1(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link="log")
dST1(x, mu = 0, sigma = 1, nu = 0, tau = 2, log = FALSE)
pST1(q, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, log.p = FALSE)
qST1(p, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, log.p = FALSE)
rST1(n, mu = 0, sigma = 1, nu = 0, tau = 2)ST2(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link = "log")
dST2(x, mu = 0, sigma = 1, nu = 0, tau = 2, log = FALSE)
pST2(q, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, log.p = FALSE)
qST2(p, mu = 1, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, log.p = FALSE)
rST2(n, mu = 0, sigma = 1, nu = 0, tau = 2)
ST3(mu.link = "identity", sigma.link = "log", nu.link = "log", tau.link = "log")
dST3(x, mu = 0, sigma = 1, nu = 1, tau = 10, log = FALSE)
pST3(q, mu = 0, sigma = 1, nu = 1, tau = 10, lower.tail = TRUE, log.p = FALSE)
qST3(p, mu = 0, sigma = 1, nu = 1, tau = 10, lower.tail = TRUE, log.p = FALSE)
rST3(n, mu = 0, sigma = 1, nu = 1, tau = 10)
SST(mu.link = "identity", sigma.link = "log", nu.link = "log",
tau.link = "logshiftto2")
dSST(x, mu = 0, sigma = 1, nu = 0.8, tau = 7, log = FALSE)
pSST(q, mu = 0, sigma = 1, nu = 0.8, tau = 7, lower.tail = TRUE, log.p = FALSE)
qSST(p, mu = 0, sigma = 1, nu = 0.8, tau = 7, lower.tail = TRUE, log.p = FALSE)
rSST(n, mu = 0, sigma = 1, nu = 0.8, tau = 7)
ST4(mu.link = "identity", sigma.link = "log", nu.link = "log", tau.link = "log")
dST4(x, mu = 0, sigma = 1, nu = 1, tau = 10, log = FALSE)
pST4(q, mu = 0, sigma = 1, nu = 1, tau = 10, lower.tail = TRUE, log.p = FALSE)
qST4(p, mu = 0, sigma = 1, nu = 1, tau = 10, lower.tail = TRUE, log.p = FALSE)
rST4(n, mu = 0, sigma = 1, nu = 1, tau = 10)
ST5(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link = "log")
dST5(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)
pST5(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)
qST5(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)
rST5(n, mu = 0, sigma = 1, nu = 0, tau = 1)
mu.link, with "identity" link as the default for the mu parameter.
Other links are "$1/mu^2$" and "log"sigma.link, with "log" link as the default for the sigma parameter.
Other links are "inverse" and "identity"nu.link, with "identity" link as the default for the nu parameter.
Other links are "$1/mu^2$" and "log"nu.link, with "log" link as the default for the nu parameter.
Other links are "inverse", "identity"mu parameter valuesnu parameter valuestau parameter valueslength(n) > 1, the length is
taken to be the number requiredST1(), ST2(), ST3(), ST4() and ST5() return a gamlss.family object
which can be used to fit the skew t type 1-5 distribution in the gamlss() function.
dST1(), dST2(), dST3(), dST4() and dST5() give the density functions,
pST1(), pST2(), pST3(), pST4() and pST5() give the cumulative distribution functions,
qST1(), qST2(), qST3(), qST4() and qST5() give the quantile function, and
rST1(), rST2(), rST3(), rST4() and rST3() generates random deviates.ST1), Azzalini (1986) is defined as
$$f(y|\mu,\sigma,\nu, \tau)=\frac{z}{\sigma} \mbox{\hspace{0.1cm}}f_z(z) \mbox{\hspace{0.1cm}} F_z(\nu z)$$for $-\infty
The probability density function of the skew t distribution type 2, (ST2), Azzalini and Capitano (2003),
is defined as
$$f(y|\mu,\sigma,\nu,\frac{z}{\sigma} \mbox{\hspace{0.1cm}} f_{z_1}(z) \mbox{\hspace{0.1cm}} F_{z_2}(w) \tau)=$$
for $-\infty The probability density function of the skew t distribution type q, ( The probability density function of the skew t distribution type q, (
The probability density function of the skew t distribution type 5, ( where $c=2^{a +b-1} (a+b)^{1/2} B(a,b)$, and
$B(a,b)=\Gamma(a)\Gamma(b)/ \Gamma(a+b)$ and
$z=(y-\mu)/\sigma$ and
$\nu=(a-b)/\left[ab(a+b) \right]^{1/2}$
and
$\tau=2/(a+b)$
for $-\inftyST3), is defined in Chapter 10 of the
GAMLSS manual.
ST4), is defined in Chapter of the
GAMLSS manual. ST5), is defined as
$$f(y|\mu,\sigma,\nu, \tau)=\frac{1}{c} \left[ 1+ \frac{z}{(a+b +z^2)^{1/2}} \right]^{a+1/2} \left[ 1- \frac{z}{(a+b+z^2)^{1/2}}\right]^{b+1/2}$$
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.
Stasinopoulos D. M. Rigby R. A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R.
Accompanying documentation in the current GAMLSS help files, (see also
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,
gamlss.family, BCCG, GA,
IG LNOy<- rST5(200, mu=5, sigma=1, nu=.1)
hist(y)
curve(dST5(x, mu=30 ,sigma=5,nu=-1), -50, 50, main = "The ST5 density mu=30 ,sigma=5,nu=1")
# library(gamlss)
# m1<-gamlss(y~1, family=ST1)
# m2<-gamlss(y~1, family=ST2)
# m3<-gamlss(y~1, family=ST3)
# m4<-gamlss(y~1, family=ST4)
# m5<-gamlss(y~1, family=ST5)
# GAIC(m1,m2,m3,m4,m5)Run the code above in your browser using DataLab