The skew t distributions, type 1 to 5
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.
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)
mu.link |
Defines the |
sigma.link |
Defines the |
nu.link |
Defines the |
tau.link |
Defines the |
x,q |
vector of quantiles |
mu |
vector of |
sigma |
vector of scale parameter values |
nu |
vector of |
tau |
vector of |
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 |
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.
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.
The mean of the ex-Gaussian is mu+nu and the variance is sigma^2+nu^2.
Bob Rigby and Mikis Stasinopoulos
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.
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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