SN1 {gamlss.dist}

Skew Normal Type 1 distribution for fitting a GAMLSS

Description

The function SN1() defines the Skew Normal Type 1 distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(), with parameters mu, sigma and nu. The functions dSN1, pSN1, qSN1 and rSN1 define the density, distribution function, quantile function and random generation for the SN1 parameterization of the Skew Normal Type 1 distribution.

Usage

SN1(mu.link = "identity", sigma.link = "log", nu.link="identity")
dSN1(x, mu = 0, sigma = 1, nu = 0, log = FALSE)
pSN1(q, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)
qSN1(p, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)
rSN1(n, mu = 0, sigma = 1, nu = 0)

Arguments

mu.link

Defines the mu.link, with "‘identity"’ links the default for the mu parameter

sigma.link

Defines the sigma.link, with "‘log"’ as the default for the sigma parameter

nu.link

Defines the nu.link, with "‘identity"’ as the default for the sigma parameter

x, q

vector of quantiles

mu

vector of location parameter values

sigma

vector of scale parameter values

nu

vector of scale 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 parameterization of the Skew Normal Type 1 distribution in the function SN1 is

f(y|\mu,\sigma,\nu)=\frac{2}{\sigma} \phi(z) \Phi(\nu z)

for (-\infty<y<+\infty), (-\infty<\mu<+\infty), \sigma>0 and (-\infty<\nu<+\infty) where z=(y-\mu)/ \sigma and \phi() and \Phi() are the pdf and cdf of the standard nornal distribution, respectively, see pp. 378-379 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a Skew Normal Type 1 distribution in the gamlss() function.

Note

This is a special case of the Skew Exponential Power type 1 distribution (SEP1) where tau=2.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Fiona McElduff

References

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, 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, 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. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

par(mfrow=c(2,2))
y<-seq(-3,3,0.2)
plot(y, dSN1(y), type="l" , lwd=2)
q<-seq(-3,3,0.2)
plot(q, pSN1(q), ylim=c(0,1), type="l", lwd=2) 
p<-seq(0.0001,0.999,0.05)
plot(p, qSN1(p), type="l", lwd=2)
dat <- rSN1(100)
hist(rSN1(100), nclass=30)