# Zero inflated and zero adjusted Binomial distribution for fitting in GAMLSS

### 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 |

`sigma.link` |
Defines the |

`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/)

### See Also

`gamlss.family`

, `BI`

### Examples

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')
``` |