NO2: Normal distribution (with variance as sigma parameter) for fitting a GAMLSS

Description

The function NO2() 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 mu and variance equal to sigma. The functions dNO2, pNO2, qNO2 and rNO2 define the density, distribution function, quantile function and random generation for this specific parameterization of the normal distribution.

[A alternative parameterization with sigma as the standard deviation is given in the function NO()]

Usage

NO2(mu.link = "identity", sigma.link = "log")
dNO2(x, mu = 0, sigma = 1, log = FALSE)
pNO2(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qNO2(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rNO2(n, mu = 0, sigma = 1)

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

x,q

vector of quantiles

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]

vector of probabilities.

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 NO2() is $$f(y|\mu,\sigma)=\frac{1}{\sqrt{2 \pi \sigma}}\exp\left[-\frac{1}{2}\frac{(y-\mu)^2}{\sigma}\right]$$

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.

Examples

Run this code

# NOT RUN {
NO()# gives information about the default links for the normal distribution
dat<-rNO(100)
hist(dat)        
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 
# }

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