gamlss.dist (version 1.5-0)

ST3: The skew t distribution, type 3

Description

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.

Usage

ST3(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link="log")
dST3(y, mu = 0, sigma = 1, nu = 0, tau=1, log = FALSE)
pST3(q, mu = 0, sigma = 1, nu = 0, tau=1, lower.tail = TRUE, log.p = FALSE)
qST3(p, mu = 0, sigma = 1, nu = 0, tau=1, lower.tail = TRUE, log.p = FALSE)
rST3(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"

y,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

• ST3() returns a gamlss.family object which can be used to fit the skew t type 3 distribution in the gamlss() function. dST3() gives the density, pST3() gives the distribution function, qST3() gives the quantile function, and rST3() generates random deviates.

Details

The probability density function of the skew t distribution type 3, (ST3), 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 $c=z^{a +b-1} (a+b)^{1/2} B(a,b)$, 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 $-\infty\infty$, $-\infty<\mu>\infty$, $\sigma>0$, $-\infty<\nu>\infty$ and $\tau>0$.

References

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. 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.londonmet.ac.uk/gamlss/).

See Also

gamlss, gamlss.family, BCCG, GA, IG LNO

Examples

y<- rST3(100, mu=10, nu=1, sigma=3)
hist(y)
m1<-gamlss(y~1, family=ST3) 
plot(m1)
curve(dST3(y=x, mu=300 ,sigma=35,nu=100), 100, 600, 
 main = "The ex-GAUS  density mu=300 ,sigma=35,nu=100")
plot(function(x) pST3(x, mu=300,sigma=35,nu=100), 100, 600, 
 main = "The ex-GAUS  cdf mu=300, sigma=35, nu=100")