gamlss.dist (version 5.3-2)

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

Value

The functions 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.

The functions dST1(), dST2(), dST3(), dST4() and dST5() give the density functions.

The funcions pST1(), pST2(), pST3(), pST4() and pST5() give the cumulative distribution functions.

The 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 \(-\infty<y<\infty\), where \(z=(y-\mu)/\sigma\), \(w=\nu \lambda^{1/2}z\), \(\lambda=(\tau+1)/(\tau+z^2)\) and \(z_1 \sim TF(0,1,\tau)\) and \(z_2 \sim TF(0,1, \tau+1)\).

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 $$f(y|\mu,\sigma,\nu, \tau)=\frac{1}{c} \left[ 1+ \frac{z}{(a+b +z^2)^{1/2}} \right]^{a+1/2} \left[ 1- \frac{z}{(a+b+z^2)^{1/2}}\right]^{b+1/2}$$

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 \(-\infty<y<\infty\), \(-\infty<\mu<\infty\), \(\sigma>0\), \(-\infty<\nu>\infty\) 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.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

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, https://www.jstatsoft.org/v23/i07.

See Also

gamlss.family, SEP1, SHASH

Examples

# NOT RUN {
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)
# }