SHASH: The Sinh-Arcsinh (SHASH) distribution for fitting a GAMLSS In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ```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)=c/(sqrt(2*pi)*sigma*(1+z^2)^(1/2)) exp(-(r^2)/2)

where

r=0.5*[exp(tau*sinh^(-1)(z))-exp(-nu*sinh^(-1)(z))]

and

c=0.5*[rho*exp(tau*sinh^(-1)(z))+ nu * exp(-nu*sinh^(-1)(z))]

and z=(y-mu)/sigma for 0<y<0, mu=(-Inf,+Inf), 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 ν<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)= (tau/sigma)* (c/sqrt(2*pi)) 1/(2*(1+z^2)^(1/2)) exp(-(r^2/2))

where

r= sinh(tau*asinh(z)-nu)

and

c= cosh(τ asinh(z)-ν)

and z=(y-mu)/sigma for 0<y<0, mu=(-Inf,+Inf), sigma>0, nu=(-Inf,+Inf) 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|μ,σ,ν,τ)= (c/sigma)*(tau/((2 π)^(1/2)))*(1/((1+z^2)^(1/2)))-exp(-r^2*0.5)

where z=(y-mu)/(sigma*tau), with r and c as for the pdf of the `SHASHo` distribution, for 0<y<0, mu=(-Inf,+Inf), sigma>0, nu=(-Inf,+Inf) 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 [email protected] 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.

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.

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.

`gamlss.family`, `JSU`, `BCT`

Examples

 ```1 2 3 4 5 6 7 8 9``` ```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)) ```

Example output

```Loading required package: MASS

GAMLSS Family: SHASH Sinh-Arcsinh
Link function for mu   : identity