# ZABI: Zero inflated and zero adjusted Binomial distribution for... In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The `ZABI()` function defines the zero adjusted binomial distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dZABI`, `pZABI`, `qZABI` and `rZABI` define the density, distribution function, quantile function and random generation for the zero adjusted binomial, `ZABI()`, distribution.

The `ZIBI()` function defines the zero inflated binomial distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dZIBI`, `pZIBI`, `qZIBI` and `rZIBI` define the density, distribution function, quantile function and random generation for the zero inflated binomial, `ZIBI()`, distribution.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ``` ZABI(mu.link = "logit", sigma.link = "logit") dZABI(x, bd = 1, mu = 0.5, sigma = 0.1, log = FALSE) pZABI(q, bd = 1, mu = 0.5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE) qZABI(p, bd = 1, mu = 0.5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE) rZABI(n, bd = 1, mu = 0.5, sigma = 0.1) ZIBI(mu.link = "logit", sigma.link = "logit") dZIBI(x, bd = 1, mu = 0.5, sigma = 0.1, log = FALSE) pZIBI(q, bd = 1, mu = 0.5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE) qZIBI(p, bd = 1, mu = 0.5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE) rZIBI(n, bd = 1, mu = 0.5, sigma = 0.1) ```

## Arguments

 `mu.link` Defines the `mu.link`, with "logit" link as the default for the `mu` parameter. Other links are "probit" and "cloglog"'(complementary log-log) `sigma.link` Defines the `sigma.link`, with "logit" link as the default for the `mu` parameter. Other links are "probit" and "cloglog"'(complementary log-log) `x` vector of (non-negative integer) quantiles `mu` vector of positive probabilities `sigma` vector of positive probabilities `bd` vector of binomial denominators `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

For the definition of the distributions see Rigby and Stasinopoulos (2010) below.

## Value

The functions `ZABI` and `ZIBI` return a `gamlss.family` object which can be used to fit a binomial distribution in the `gamlss()` function.

## Note

The response variable should be a matrix containing two columns, the first with the count of successes and the second with the count of failures.

## Author(s)

Mikis Stasinopoulos, 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`, `BI`
 ```1 2 3 4 5 6 7 8 9``` ```ZABI() curve(dZABI(x, mu = .5, bd=10), from=0, to=10, n=10+1, type="h") tN <- table(Ni <- rZABI(1000, mu=.2, sigma=.3, bd=10)) r <- barplot(tN, col='lightblue') ZIBI() curve(dZIBI(x, mu = .5, bd=10), from=0, to=10, n=10+1, type="h") tN <- table(Ni <- rZIBI(1000, mu=.2, sigma=.3, bd=10)) r <- barplot(tN, col='lightblue') ```