NO: Normal distribution for fitting a GAMLSS

NOR Documentation

Normal distribution for fitting a GAMLSS


The function NO() defines the normal distribution, a two parameter distribution, for a object to be used in GAMLSS fitting using the function gamlss(), with mean equal to the parameter mu and sigma equal the standard deviation. The functions dNO, pNO, qNO and rNO define the density, distribution function, quantile function and random generation for the NO parameterization of the normal distribution. [A alternative parameterization with sigma equal to the variance is given in the function NO2()]


NO( = "identity", = "log")
dNO(x, mu = 0, sigma = 1, log = FALSE)
pNO(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qNO(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rNO(n, mu = 0, sigma = 1)


Defines the, with "identity" link as the default for the mu parameter

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


vector of quantiles


vector of location parameter values


vector of scale parameter values

log, log.p

logical; if TRUE, probabilities p are given as log(p).


logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]


vector of probabilities.


number of observations. If length(n) > 1, the length is taken to be the number required


The parametrization of the normal distribution given in the function NO() is

f(y|\mu,\sigma)=\frac{1}{\sqrt{2 \pi }\sigma}\exp \left[-\frac{1}{2}(\frac{y-\mu}{\sigma})^2\right]

for y=(-\infty,\infty), \mu=(-\infty,+\infty) and \sigma>0 see pp. 369-370 of Rigby et al. (2019).


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


For the function NO(), \mu is the mean and \sigma is the standard deviation (not the variance) of the normal distribution.


Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou


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

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

See Also, NO2


NO()# gives information about the default links for the normal distribution
plot(function(y) dNO(y, mu=10 ,sigma=2), 0, 20)
plot(function(y) pNO(y, mu=10 ,sigma=2), 0, 20)
plot(function(y) qNO(y, mu=10 ,sigma=2), 0, 1)
# library(gamlss)        
# gamlss(dat~1,family=NO) # fits a constant for mu and sigma 

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.