R: Normal distribution (with variance as sigma parameter) for...

NO2 {gamlss.dist} | R Documentation

## 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
`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

### Details

The parametrization of the normal distribution given in the function `NO2()` is

f(y|mu,sigma)=(1/(sqrt(2*pi*sigma))* exp(-0.5*(y-mu)^2/sigma)

for y=(-Inf,+Inf), μ=(-Inf,+Inf) and σ>0.

### Value

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

### Note

For the function `NO()`, mu is the mean and sigma is the standard deviation (not the variance) of the normal distribution. [The function `NO2()` defines the normal distribution with sigma as the variance.]

### Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

### 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.

### See Also

`gamlss.family`, `NO`

### Examples

    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

[Package gamlss.dist version 5.3-2 Index]