NOF: Normal distribution family for fitting a GAMLSS

In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

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 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. This is an instance of the Taylor's power low, see Enki et al. (2017). 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

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) see pp. 373-374 of Rigby et al. (2019).

Value

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

Note

For the function NOF(), \mu is the mean and \sigma \mu^{\nu/2} is the standard deviation of the normal distribution family. The NOF is design for fitting regression type models where the variance is proportional to a power of the mean. Models of this type are also related to the "pseudo likelihood" models of Carroll and Rubert (1987) but here a proper likelihood is miximised.

Note that because the high correlation between the sigma and the nu parameter the mixed() method should be used in the fitting.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

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

Enki, D G, Noufaily, A., Farrington, P., Garthwaite, P., Andrews, N. and Charlett, A. (2017) Taylor's power law and the statistical modelling of infectious disease surveillance data, Journal of the Royal Statistical Society: Series A (Statistics in Society), volume=180, number=1, pages=45-72.

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, NO, NO2

Examples

NOF()# gives information about the default links for the normal distribution family
## Not run: 
## the normal distribution, fitting a constant sigma
m1<-gamlss(y~poly(x,2), sigma.fo=~1, family=NO, data=abdom)
## the normal family, fitting a variance proportional to the mean (mu)
m2<-gamlss(y~poly(x,2), sigma.fo=~1, family=NOF, data=abdom, method=mixed(1,20))
## the nornal distribution fitting  the 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)

## End(Not run)

mstasinopoulos/GAMLSS-Distibutions documentation built on Nov. 3, 2023, 10:33 a.m.