# ZANBI: Zero inflated and zero adjusted negative binomial... In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The function `ZINBI` defines the zero inflated negative binomial distribution, a three parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dZINBI`, `pZINBI`, `qZINBI` and `rZINBI` define the density, distribution function, quantile function and random generation for the zero inflated negative binomial, `ZINBI()`, distribution.

The function `ZANBI` defines the zero adjusted negative binomial distribution, a three parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dZANBI`, `pZANBI`, `qZANBI` and `rZANBI` define the density, distribution function, quantile function and random generation for the zero inflated negative binomial, `ZANBI()`, distribution.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```ZINBI(mu.link = "log", sigma.link = "log", nu.link = "logit") dZINBI(x, mu = 1, sigma = 1, nu = 0.3, log = FALSE) pZINBI(q, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE) qZINBI(p, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE) rZINBI(n, mu = 1, sigma = 1, nu = 0.3) ZANBI(mu.link = "log", sigma.link = "log", nu.link = "logit") dZANBI(x, mu = 1, sigma = 1, nu = 0.3, log = FALSE) pZANBI(q, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE) qZANBI(p, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE) rZANBI(n, mu = 1, sigma = 1, nu = 0.3) ```

## 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 `nu.link` Defines the `mu.link`, with "logit" link as the default for the nu parameter `x` vector of (non-negative integer) quantiles `mu` vector of positive means `sigma` vector of positive despersion parameter `nu` vector of zero probability 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

The definition for the zero inflated Negative Binomial type I distribution and for the zero adjusted Negative Binomial type I distribution is given in Rigby and Stasinopoulos (2010) below

## Value

The functions `ZINBI` and `ZANBI` return a `gamlss.family` object which can be used to fit a zero inflated or zero adjusted Negative Binomial type I distribution respectively in the `gamlss()` function.

## Author(s)

Mikis Stasinopoulos [email protected], Bob Rigby

## References

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.

Rigby, R. A. and Stasinopoulos D. M. (2010) The gamlss.family distributions, (distributed with this package or see http://www.gamlss.org/)

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`, `NBI`, `NBII`
 ```1 2 3 4 5 6 7``` ```ZINBI() ZANBI() # creating data and plotting them dat <- rZINBI(1000, mu=5, sigma=.5, nu=0.1) r <- barplot(table(dat), col='lightblue') dat1 <- rZANBI(1000, mu=5, sigma=.5, nu=0.1) r1 <- barplot(table(dat1), col='lightblue') ```