R: Binomial distribution for fitting a GAMLSS

BI {gamlss.dist} | R Documentation

## Binomial distribution for fitting a GAMLSS

### 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

    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.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

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, https://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.

### See Also

`gamlss.family`, `ZABI`, `ZIBI`

### Examples

     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')

[Package gamlss.dist version 5.3-2 Index]