R: Normal distribution family for fitting a GAMLSS

NOF {gamlss.dist} | R Documentation

## 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)=(1/(sqrt(2*pi)*sigma*mu^(nu/2)))* exp(-0.5*((y-mu)^2/sigma^2*mu^nu))

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

### 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.

RRigby, 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. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also 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`, `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)

[Package gamlss.dist version 5.3-2 Index]