R: Negative Binomial type I distribution for fitting a GAMLSS

NBI {gamlss.dist} | R Documentation

## Negative Binomial type I distribution for fitting a GAMLSS

### Description

The `NBI()` function defines the Negative Binomial type I distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dNBI`, `pNBI`, `qNBI` and `rNBI` define the density, distribution function, quantile function and random generation for the Negative Binomial type I, `NBI()`, distribution.

### Usage

    NBI(mu.link = "log", sigma.link = "log")
    dNBI(x, mu = 1, sigma = 1, log = FALSE)
    pNBI(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    qNBI(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
    rNBI(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 I distribution.

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

for y=0,1,2, ...,Inf, μ>0 and σ>0. This parameterization is equivalent to that used by Anscombe (1950) except he used alpha=1/sigma instead of sigma.

### Value

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

### Warning

For values of sigma<0.0001 the d,p,q,r functions switch to the Poisson distribution

### Note

mu is the mean and (mu+sigma*mu^2)^0.5 is the standard deviation of the Negative Binomial type I distribution (so sigma is the dispersion parameter in the usual GLM for the negative binomial type I distribution)

### Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

### References

Anscombe, F. J. (1950) Sampling theory of the negative bimomial and logarithmic distributiona, 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. (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`, `NBII`, `PIG`, `SI`

### Examples

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

[Package gamlss.dist version 5.3-2 Index]