SHASH: The Sinh-Arcsinh (SHASH) distribution for fitting a GAMLSS
In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape
SHASH R Documentation
The Sinh-Arcsinh (SHASH) distribution for fitting a GAMLSS
Description
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.
Usage
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)
Arguments
mu.link
Defines the mu.link, with "identity" link as the default for the mu parameter.
sigma.link
Defines the sigma.link, with "log" link as the default for the sigma parameter.
nu.link
Defines the nu.link, with "log" link as the default for the nu parameter.
tau.link
Defines the tau.link, with "log" link as the default for the tau parameter.
x,q
vector of quantiles
mu
vector of location parameter values
sigma
vector of scale parameter values
nu
vector of skewness nu parameter values
tau
vector of kurtosis 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
Details
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 ,
-\infty<\mu<\infty,
\sigma>0,
\nu>0 and
\tau>0, see pp. 396-397 of Rigby et al. (2019).
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 c}{\sigma \sqrt{2 \pi} (1+z^2)^{1/2}} \exp{(-\frac{1}{2} r^2)}
where
r= \sinh(\tau \, \sinh^{-1}(z)-\nu)
and
c= \cosh(\tau \sinh^{-1}(z)-\nu)
and z=(y-\mu)/\sigma
for -\infty < y < \infty ,
-\infty<\mu<+\infty,
\sigma>0,
-\infty<\nu<+\infty and
\tau>0, see pp. 398-400 of Rigby et al. (2019)
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.
Value
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.
Warning
The qSHASH and rSHASH are slow since they are relying on golden section for finding the quantiles
Author(s)
Bob Rigby, Mikis Stasinopoulos and Fiona McElduff
References
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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9780429298547")}. 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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v023.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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}
(see also https://www.gamlss.com/).
See Also
gamlss.family, JSU, BCT
Examples
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))
gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.
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| SHASH | R Documentation |
The Sinh-Arcsinh (SHASH) distribution for fitting a GAMLSS
Description
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.
Usage
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)
Arguments
mu.link |
Defines the |
sigma.link |
Defines the |
nu.link |
Defines the |
tau.link |
Defines the |
x,q |
vector of quantiles |
mu |
vector of location parameter values |
sigma |
vector of scale parameter values |
nu |
vector of skewness |
tau |
vector of kurtosis |
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 |
Details
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 ,
-\infty<\mu<\infty,
\sigma>0,
\nu>0 and
\tau>0, see pp. 396-397 of Rigby et al. (2019).
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 c}{\sigma \sqrt{2 \pi} (1+z^2)^{1/2}} \exp{(-\frac{1}{2} r^2)}
where
r= \sinh(\tau \, \sinh^{-1}(z)-\nu)
and
c= \cosh(\tau \sinh^{-1}(z)-\nu)
and z=(y-\mu)/\sigma
for -\infty < y < \infty ,
-\infty<\mu<+\infty,
\sigma>0,
-\infty<\nu<+\infty and
\tau>0, see pp. 398-400 of Rigby et al. (2019)
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.
Value
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.
Warning
The qSHASH and rSHASH are slow since they are relying on golden section for finding the quantiles
Author(s)
Bob Rigby, Mikis Stasinopoulos and Fiona McElduff
References
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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9780429298547")}. 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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v023.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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}
(see also https://www.gamlss.com/).
See Also
gamlss.family, JSU, BCT
Examples
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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