gamlss.dist (version 5.0-0)

NOF: Normal distribution family for fitting a GAMLSS

Description

The function NOF() defines a normal distribution family, which has three parameters. The distribution can be used in a GAMLSS fitting using the function gamlss(). The mean of NOF is equal to mu. The variance is equal to sigma^2*mu^nu so the standard deviation is sigma*mu^(nu/2). The function is design for cases where the variance is proportional to a power of the mean. The functions dNOF, pNOF, qNOF and rNOF define the density, distribution function, quantile function and random generation for the NOF parametrization of the normal distribution family.

Usage

NOF(mu.link = "identity", sigma.link = "log", nu.link = "identity")
dNOF(x, mu = 0, sigma = 1, nu = 0, log = FALSE)
pNOF(q, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)
qNOF(p, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)
rNOF(n, mu = 0, sigma = 1, nu = 0)

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 "identity" link as the default for the nu parameter

x,q

vector of quantiles

mu

vector of location parameter values

sigma

vector of scale parameter values

nu

vector of power 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

Value

returns a gamlss.family object which can be used to fit a normal distribution family in the gamlss() function.

Details

The parametrization of the normal distribution given in the function NOF() is $$f(y|\mu,\sigma, \nu)=\frac{1}{\sqrt{2 \pi }\sigma \mu^{\nu/2}}\exp \left[-\frac{1}{2}\frac{(y-\mu)^2}{\sigma^2 \mu^\nu}\right]$$

for \(y=(-\infty,\infty)\), \(\mu=(-\infty,\infty)\), \(\sigma>0\) and \(\nu=(-\infty,+\infty)\).

References

Davidian, M. and Carroll, R. J. (1987), Variance Function Estimation, Journal of the American Statistical Association, Vol. 82, pp. 1079-1091

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.

See Also

gamlss.family, NO, NO2

Examples

NOF()# gives information about the default links for the normal distribution family
# library(gamlss)
#data(abdom)        
## the normal distribution fit with constant sigma
#m1<-gamlss(y~poly(x,2), sigma.fo=~1, family=NO, data=abdom)
## the normal family fit with variance proportional to mu
#m2<-gamlss(y~poly(x,2), sigma.fo=~1, family=NOF, data=abdom, method=mixed(1,20))
## a nornal distribution fit with variance as a function of x
#m3 <-gamlss(y~poly(x,2), sigma.fo=~x,   family=NO, data=abdom, method=mixed(1,20)) 
#GAIC(m1,m2,m3)