R: Negative Binomial type II distribution for fitting a GAMLSS

NBII {gamlss.dist} | R Documentation

## Negative Binomial type II distribution for fitting a GAMLSS

### Description

The `NBII()` function defines the Negative Binomial type II distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dNBII`, `pNBII`, `qNBII` and `rNBII` define the density, distribution function, quantile function and random generation for the Negative Binomial type II, `NBII()`, distribution.

### Usage

    NBII(mu.link = "log", sigma.link = "log")
    dNBII(x, mu = 1, sigma = 1, log = FALSE)
    pNBII(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    qNBII(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    rNBII(n, mu = 1, sigma = 1)

### Arguments

`mu.link` | Defines the `mu.link`, with "log" 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` | vector of (non-negative integer) quantiles
`mu` | vector of positive means
`sigma` | vector of positive despersion parameter
`p` | vector of probabilities
`q` | vector of quantiles
`n` | number of random values to return
`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]

### Details

Definition file for Negative Binomial type II distribution.

P(Y=y|μ,σ)= Γ(y+μ/σ) σ^y / Γ(μ/σ)Γ(y+1) (1+σ)^{y+μ/σ}

for y=0,1,2, ...,Inf, μ>0 and σ>0. This parameterization was used by Evans (1953) and also by Johnson et al. (1993) p 200.

### Value

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

### Note

mu is the mean and ((1+sigma)*mu)**0.5 is the standard deviation of the Negative Binomial type II distribution, so sigma is a dispersion parameter

### Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

### References

Evans, D. A. (1953). Experimental evidence concerning contagious distributions in ecology. Biometrika, 40: 186-211.

Johnson, N. L., Kotz, S. and Kemp, A. W. (1993). Univariate Discrete Distributions, 2nd edn. Wiley, New York.

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.

### See Also

`gamlss.family`, `NBI`, `PIG`, `SI`

### Examples

    NBII()  # gives information about the default links for the Negative Binomial type II distribution
    # plotting the distribution
    plot(function(y) dNBII(y, mu = 10, sigma = 0.5 ), from=0, to=40, n=40+1, type="h")
    # creating random variables and plot them
    tN <- table(Ni <- rNBII(1000, mu=5, sigma=0.5))
    r <- barplot(tN, col='lightblue')
    # library(gamlss)
    # data(aids)
    # h<-gamlss(y~cs(x,df=7)+qrt, family=NBII, data=aids) # fits a model
    # plot(h)
    # pdf.plot(family=NBII, mu=10, sigma=0.5, min=0, max=40, step=1)

[Package gamlss.dist version 5.3-2 Index]