ST1: The skew t distributions, type 1 to 5

In Stan125/gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

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

Details

f(y|mu,sigma,nu,tau)=z/sigma f_z1(z)F_z2(w)

for -Inf<y<Inf, where z=(y-mu)/sigma, w=nu*sqrt(lambda)*z, lambda=(tau+1)/(tau+z*z) and z_1 ~ TF(01,1,tau) and z_2 ~ 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)=(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=2^(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.

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.

Note

The mean of the ex-Gaussian is mu+nu and the variance is sigma^2+nu^2.

Author(s)

Bob Rigby and Mikis Stasinopoulos

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

 
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)

Stan125/gamlss.dist documentation built on May 12, 2019, 7:38 a.m.