gamlss.dist (version 5.3-2)

# NO: Normal distribution for fitting a GAMLSS

## Description

The function NO() defines the normal distribution, a two parameter distribution, for a gamlss.family 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()]

## Usage

NO(mu.link = "identity", sigma.link = "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)

## Arguments

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

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

x,q

vector of quantiles

mu

vector of location parameter values

sigma

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

## Value

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

## Details

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 $$\sigma>0$$.

## 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. 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, https://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, NO2

## Examples

# NOT RUN {