# BI: Binomial distribution for fitting a GAMLSS In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The `BI()` function defines the binomial distribution, a one parameter family distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dBI`, `pBI`, `qBI` and `rBI` define the density, distribution function, quantile function and random generation for the binomial, `BI()`, distribution.

## Usage

 ```1 2 3 4 5``` ```BI(mu.link = "logit") dBI(x, bd = 1, mu = 0.5, log = FALSE) pBI(q, bd = 1, mu = 0.5, lower.tail = TRUE, log.p = FALSE) qBI(p, bd = 1, mu = 0.5, lower.tail = TRUE, log.p = FALSE) rBI(n, bd = 1, mu = 0.5) ```

## 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) `x` vector of (non-negative integer) quantiles `mu` 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

Definition file for binomial distribution.

f(y|mu)=(Gamma(n+1)*Gamma(y+1)/Gamma(n-y+1))* mu^y *(1-mu)^(n-y)

for y=0,1,2,...,n and 0<μ< 1.

## Value

returns 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. The parameter `mu` represents a probability parameter with limits 0 < mu <1. n*mu is the mean of the distribution where n is the binomial denominator.

## Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

## 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.

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`, `ZABI`, `ZIBI`

## Examples

 ```1 2 3 4 5 6 7 8``` ``` BI()# gives information about the default links for the Binomial distribution # data(aep) # library(gamlss) # h<-gamlss(y~ward+loglos+year, family=BI, data=aep) # plot of the binomial distribution curve(dBI(x, mu = .5, bd=10), from=0, to=10, n=10+1, type="h") tN <- table(Ni <- rBI(1000, mu=.2, bd=10)) r <- barplot(tN, col='lightblue') ```

### Example output

```Loading required package: MASS

GAMLSS Family: BI Binomial
Link function for mu   : logit
```

gamlss.dist documentation built on June 6, 2018, 5:04 p.m.