There are 5 different skew t distributions implemented in GAMLSS.
The Skew t type 1 distribution, ST1, is based on Azzalini (1986).
The skew t type 2 distribution, ST2, is based on Azzalini and Capitanio (2003).
The skew t type 3 , ST3 and ST3C, distribution is based Fernande and Steel (1998).
The difference betwwen the ST3 and ST3C is that the first is written entirely in R while
the second is in C.
The skew t type 4 distribution , ST4, is a spliced-shape distribution.
The skew t type 5 distribution , ST5, is Jones and Faddy (2003).
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)
ST3C(mu.link = "identity", sigma.link = "log", nu.link = "log", tau.link = "log")
dST3C(x, mu = 0, sigma = 1, nu = 1, tau = 10, log = FALSE)
pST3C(q, mu = 0, sigma = 1, nu = 1, tau = 10, lower.tail = TRUE, log.p = FALSE)
qST3C(p, mu = 0, sigma = 1, nu = 1, tau = 10, lower.tail = TRUE, log.p = FALSE)
rST3C(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)
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 "\(1/mu^2\)" and "log"
Defines the nu.link, with "log" link as the default for the nu parameter.
Other links are "inverse", "identity"
vector of quantiles
vector of mu parameter values
vector of scale parameter values
vector of nu parameter values
vector of 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
for extra arguments
ST1(), 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.
$$f(y|\mu,\sigma,\nu,\frac{z}{\sigma} \mbox{\hspace{0.1cm}} f_{z_1}(z) \mbox{\hspace{0.1cm}} F_{z_2}(w) \tau)=$$
for \(-\infty<y<\infty\), where \(z=(y-\mu)/\sigma\), \(w=\nu \lambda^{1/2}z\), \(\lambda=(\tau+1)/(\tau+z^2)\) and \(z_1 \sim TF(0,1,\tau)\) and \(z_2 \sim TF(0,1, \tau+1)\).
The probability density function of the skew t distribution type q, (ST3), is defined in Chapter 10 of the
GAMLSS manual.
The probability density function of the skew t distribution type q, (ST4), is defined in Chapter of the
GAMLSS manual.
The probability density function of the skew t distribution type 5, (ST5), is defined as
$$f(y|\mu,\sigma,\nu, \tau)=\frac{1}{c} \left[ 1+ \frac{z}{(a+b +z^2)^{1/2}} \right]^{a+1/2} \left[ 1- \frac{z}{(a+b+z^2)^{1/2}}\right]^{b+1/2}$$
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 \(-\infty<y<\infty\), \(-\infty<\mu<\infty\), \(\sigma>0\), \(-\infty<\nu>\infty\) and \(\tau>0\).
Azzalini A. (1986) Futher results on a class of distributions which includes the normal ones, Statistica, 46, pp. 199-208.
Azzalini A. and Capitanio, A. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65, pp. 367-389.
Jones, M.C. and Faddy, M. J. (2003) A skew extension of the t distribution, with applications. Journal of the Royal Statistical Society, Series B, 65, pp 159-174.
Fernandez, C. and Steel, M. F. (1998) On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 93, pp. 359-371.
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 http://www.gamlss.org/).
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.
y<- 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)
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