# NBI: Negative Binomial type I distribution for fitting a GAMLSS In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## 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

 ```1 2 3 4 5``` ```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 [email protected], 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.

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 http://www.gamlss.org/).

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, http://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.

`gamlss.family`, `NBII`, `PIG`, `SI`
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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) ```