R: Negative Binomial Family distribution for fitting a GAMLSS

NBF {gamlss.dist} | R Documentation

## Negative Binomial Family distribution for fitting a GAMLSS

### Description

The `NBF()` function defines the Negative Binomial family distribution, a three parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dNBF`, `pNBF`, `qNBF` and `rNBF` define the density, distribution function, quantile function and random generation for the negative binomial family, `NBF()`, distribution.

The functions `dZINBF`, `pZINBF`, `qZINBF` and `rZINBF` define the density, distribution function, quantile function and random generation for the zero inflated negative binomial family, `ZINBF()`, distribution a four parameter distribution.

### Usage

    NBF(mu.link = "log", sigma.link = "log", nu.link = "log")

    dNBF(x, mu = 1, sigma = 1, nu = 2, log = FALSE)

    pNBF(q, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

    qNBF(p, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

    rNBF(n, mu = 1, sigma = 1, nu = 2)

    ZINBF(mu.link = "log", sigma.link = "log", nu.link = "log",
          tau.link = "logit")

    dZINBF(x, mu = 1, sigma = 1, nu = 2, tau = 0.1, log = FALSE)

    pZINBF(q, mu = 1, sigma = 1, nu = 2, tau = 0.1, lower.tail = TRUE,
          log.p = FALSE)

    qZINBF(p, mu = 1, sigma = 1, nu = 2, tau = 0.1, lower.tail = TRUE,
          log.p = FALSE)

    rZINBF(n, mu = 1, sigma = 1, nu = 2, tau = 0.1)

### Arguments

`mu.link` | The link function for `mu`
`sigma.link` | The link function for `sigma`
`nu.link` | The link function for `nu`
`tau.link` | The link function for `tau`
`x` | vector of (non-negative integer)
`mu` | vector of positive means
`sigma` | vector of positive dispersion parameter
`nu` | vector of power parameter
`tau` | vector of inflation parameter
`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
`q` | vector of quantiles
`n` | number of random values to return

### Details

The definition for Negative Binomial Family distribution , `NBF`, is similar to the Negative Binomial type I. The probability function of the `NBF` can be obtained by replacing σ with σ μ^{ν-2} where ν is a power parameter. The distribution has mean μ and variance μ+σ μ^{ν}.

### Value

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

### Author(s)

Bob Rigby and Mikis Stasinopoulos

### References

Anscombe, F. J. (1950) Sampling theory of the negative binomial and logarithmic distributions, Biometrika, 37, 358-382.

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. 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.

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

`NBI`, `NBII`

### Examples

    NBF() # default link functions for the Negative Binomial Family
    # plotting the distribution
    plot(function(y) dNBF(y, mu = 10, sigma = 0.5, nu=2 ), from=0,
         to=40, n=40+1, type="h")
    # creating random variables and plot them
    tN <- table(Ni <- rNBF(1000, mu=5, sigma=0.5, nu=2))
    r <- barplot(tN, col='lightblue')
    # zero inflated NBF
    ZINBF() # default link functions  for the zero inflated NBF
    # plotting the distribution
    plot(function(y) dZINBF(y, mu = 10, sigma = 0.5, nu=2, tau=.1 ),
         from=0, to=40, n=40+1, type="h")
    # creating random variables and plot them
    tN <- table(Ni <- rZINBF(1000, mu=5, sigma=0.5, nu=2, tau=0.1))
    r <- barplot(tN, col='lightblue')
    ## Not run:
    library(gamlss)
    data(species)
    species <- transform(species, x=log(lake))
    m6 <- gamlss(fish~poly(x,2), sigma.fo=~1, data=species, family=NBF,
              n.cyc=200)
    fitted(m6, "nu")[1]

    ## End(Not run)

[Package gamlss.dist version 5.3-2 Index]