ST3 {gamlss.dist} | R Documentation |
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.
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)
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 |
The probability density function of the skew t distribution type 3, (ST3
), is defined as
f(y|mu,sigma,nu,tau)=(1/c)*(1+(z/(a+b+z^2)^0.5))^(a+0.5)*(1-(a+b+z^2)^0.5)^(b+0.5)
where c=z^(a+b-1) * (a+b)^0.5 *B(a,b), and Gamma(a)*Gamma(b)/Gamma(a+b) and (y-mu)/sigma and nu=(a-b)/(a*b*(a+b))^0.5 and tau=2/(a+b) for -Inf<y<Inf, -Inf<mu<Inf, σ>0, -Inf<nu<Inf and tau>0.
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.
The mean of the ex-Gaussian is mu+nu and the variance is sigma^2+nu^2.
Bob Rigby and Mikis Stasinopoulos
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/).
gamlss
, gamlss.family
, BCCG
, GA
, IG
LNO
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")