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gamlss.dist (version 4.3-4)

ST1: The skew t distributions, type 1 to 5

Description

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.

Usage

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)

Arguments

mu.link
Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "$1/mu^2$" and "log"
sigma.link
Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse" and "identity"
nu.link
Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "$1/mu^2$" and "log"
tau.link
Defines the nu.link, with "log" link as the default for the nu parameter. Other links are "inverse", "identity"
x,q
vector of quantiles
mu
vector of mu parameter values
sigma
vector of scale parameter values
nu
vector of nu parameter values
tau
vector of tau parameter values
log, log.p
logical; if TRUE, probabilities p are given as log(p).
lower.tail
logical; if TRUE (default), probabilities are P[X <= x],="" otherwise,="" p[x=""> x]
p
vector of probabilities.
n
number of observations. If length(n) > 1, the length is taken to be the number required
...
for extra arguments

Value

  • 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.

Details

$$f(y|\mu,\sigma,\nu,\frac{z}{\sigma} \mbox{\hspace{0.1cm}} f_{z_1}(z) \mbox{\hspace{0.1cm}} F_{z_2}(w) \tau)=$$

for $-\infty

The probability density function of the skew t distribution type q, (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 $-\infty0$, $-\infty<\nu>\infty$ and $\tau>0$.

References

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.

See Also

gamlss.family, SEP1, SHASH

Examples

Run this code
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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