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)
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.for $-\infty 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.
The probability density function of the skew t distribution type q, (ST4
), is defined in Chapter of the
GAMLSS manual. ST5
), is defined as
$$f(y|\mu,\sigma,\nu, \tau)=\frac{1}{c} \left[ 1+ \frac{z}{(a+b +z^2)^{1/2}} \right]^{a+1/2} \left[ 1- \frac{z}{(a+b+z^2)^{1/2}}\right]^{b+1/2}$$
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
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
, SEP1
, SHASH
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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