The Sinh-Arcsinh (SHASH) distribution is a four parameter distribution,
for a gamlss.family object to be used for a
GAMLSS fitting using the function gamlss(). The functions dSHASH,
pSHASH, qSHASH and rSHASH define the density,
distribution function, quantile function and random
generation for the Sinh-Arcsinh (SHASH) distribution.
There are 3 different SHASH distributions implemented in GAMLSS.
SHASH(mu.link = "identity", sigma.link = "log", nu.link = "log",
tau.link = "log")
dSHASH(x, mu = 0, sigma = 1, nu = 0.5, tau = 0.5, log = FALSE)
pSHASH(q, mu = 0, sigma = 1, nu = 0.5, tau = 0.5, lower.tail = TRUE,
log.p = FALSE)
qSHASH(p, mu = 0, sigma = 1, nu = 0.5, tau = 0.5, lower.tail = TRUE,
log.p = FALSE)
rSHASH(n, mu = 0, sigma = 1, nu = 0.5, tau = 0.5)SHASHo(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
dSHASHo(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)
pSHASHo(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE,
log.p = FALSE)
qSHASHo(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE,
log.p = FALSE)
rSHASHo(n, mu = 0, sigma = 1, nu = 0, tau = 1)
SHASHo2(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
dSHASHo2(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)
pSHASHo2(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE,
log.p = FALSE)
qSHASHo2(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE,
log.p = FALSE)
rSHASHo2(n, mu = 0, sigma = 1, nu = 0, tau = 1)
Defines the mu.link, with "identity" link as the default for the mu parameter.
Defines the sigma.link, with "log" link as the default for the sigma parameter.
Defines the nu.link, with "log" link as the default for the nu parameter.
Defines the tau.link, with "log" link as the default for the tau parameter.
vector of quantiles
vector of location parameter values
vector of scale parameter values
vector of skewness nu parameter values
vector of kurtosis tau parameter values
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
vector of probabilities.
number of observations. If length(n) > 1, the length is
taken to be the number required
SHASH() returns a gamlss.family object which can be used to fit the SHASH distribution in the gamlss() function.
dSHASH() gives the density, pSHASH() gives the distribution
function, qSHASH() gives the quantile function, and rSHASH()
generates random deviates.
The qSHASH and rSHASH are slow since they are relying on golden section for finding the quantiles
The probability density function of the Sinh-Arcsinh distribution, (SHASH), Jones(2005), is defined as
$$f(y|\mu,\sigma\,\nu,\tau) = \frac{c}{\sqrt{2 \pi} \sigma (1+z^2)^{1/2}} e^{-r^2/2}$$
where
$$r=\frac{1}{2} \left \{ \exp\left[ \tau \sinh^{-1}(z) \right] -\exp\left[ -\nu \sinh^{-1}(z) \right] \right\}$$
and
$$c=\frac{1}{2} \left \{ \tau \exp\left[ \tau \sinh^{-1}(z) \right] + \nu \exp\left[ -\nu \sinh^{-1}(z) \right] \right\}$$
and \(z=(y-\mu)/\sigma\) for \( -\infty < y < \infty \), \(\mu=(-\infty,+\infty)\), \(\sigma>0\), \(\nu>0\) and \(\tau>0\).
The parameters \(\mu\) and \(\sigma\) are the location and scale of the distribution. The parameter \(\nu\) determines the left hand tail of the distribution with \(\nu>1\) indicating a lighter tail than the normal and \(\nu<1\) heavier tail than the normal. The parameter \(\tau\) determines the right hand tail of the distribution in the same way.
The second form of the Sinh-Arcsinh distribution can be found in Jones and Pewsey (2009, p.2) denoted by SHASHo and the probability density function is defined as,
$$f(y|\mu,\sigma,\nu,\tau)= \frac{\tau}{\sigma} \frac{c}{\sqrt{2 \pi}} \frac{1}{2\,\sqrt{1+z^2}} \exp{(-\frac{r^2}{2})}$$
where
$$r= \sinh(\tau \, \arcsin(z)-\nu)$$
and
$$c= \cosh(\tau \arcsin(z)-\nu)$$
and \(z=(y-\mu)/\sigma\) for \( -\infty < y < \infty \), \(\mu=(-\infty,+\infty)\), \(\sigma>0\), \(\nu=(-\infty,+\infty)\) and \(\tau>0\).
The third form of the Sinh-Arcsinh distribution (Jones and Pewsey, 2009, p.8) divides the distribution by sigma for the density of the unstandardized variable. This distribution is denoted by SHASHo2 and has pdf
$$f(y|\mu,\sigma,\nu,\tau)= \frac{c}{\sigma} \frac{\tau}{\sqrt{2 \pi}}\frac{1}{\sqrt{1+z^2}}-\exp{-\frac{r^2}{2}}$$
where \(z=(y-\mu)/(\sigma \tau)\), with \(r\) and \(c\) as for the pdf of the SHASHo distribution,
for \( -\infty < y < \infty \),
\(\mu=(-\infty,+\infty)\),
\(\sigma>0\),
\(\nu=(-\infty,+\infty)\) and
\(\tau>0\).
Jones, M. C. (2006) p 546-547 in the discussion of Rigby, R. A. and Stasinopoulos D. M. (2005) Appl. Statist., 54, part 3.
Jones and Pewsey (2009) Sinh-arcsinh distributions. Biometrika. 96(4), pp. 761?780.
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 http://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, http://www.jstatsoft.org/v23/i07.
Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.
# NOT RUN {
SHASH() #
plot(function(x)dSHASH(x, mu=0,sigma=1, nu=1, tau=2), -5, 5,
main = "The SHASH density mu=0,sigma=1,nu=1, tau=2")
plot(function(x) pSHASH(x, mu=0,sigma=1,nu=1, tau=2), -5, 5,
main = "The BCPE cdf mu=0, sigma=1, nu=1, tau=2")
dat<-rSHASH(100,mu=10,sigma=1,nu=1,tau=1.5)
hist(dat)
# library(gamlss)
# gamlss(dat~1,family=SHASH, control=gamlss.control(n.cyc=30))
# }
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