# DBI: The Double binomial distribution In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The function `DBI()` defines the double binomial distribution, a two parameters distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dDBI`, `pDBI`, `qDBI` and `rDBI` define the density, distribution function, quantile function and random generation for the double binomial, `DBI()`, distribution. The function `GetBI_C` calculates numericaly the constant of proportionality needed for the pdf to sum up to 1.

## Usage

 ```1 2 3 4 5 6 7 8``` ```DBI(mu.link = "logit", sigma.link = "log") dDBI(x, mu = 0.5, sigma = 1, bd = 2, log = FALSE) pDBI(q, mu = 0.5, sigma = 1, bd = 2, lower.tail = TRUE, log.p = FALSE) qDBI(p, mu = 0.5, sigma = 1, bd = 2, lower.tail = TRUE, log.p = FALSE) rDBI(n, mu = 0.5, sigma = 1, bd = 2) GetBI_C(mu, sigma, bd) ```

## Arguments

 `mu.link` the link function for `mu` with default `log` `sigma.link` the link function for `sigma` with default `log` `x, q` vector of (non-negative integer) quantiles `bd` vector of binomial denominator `p` vector of probabilities `mu` the `mu` parameter `sigma` the `sigma` parameter `lower.tail` logical; if `TRUE` (default), probabilities are P[X <= x], otherwise, P[X > x] `log, log.p` logical; if `TRUE`, probabilities p are given as log(p) `n` how many random values to generate

## Details

The definition for the Double Poisson distribution first introduced by Efron (1986) is:

f(y| n, μ,σ)=[1/C(n,μ,σ)] [Γ(n+1)/Γ(y+1)Γ(n-y+1)] [y^y (n-y)^{n-y}/n^n][n^{n/σ} μ^{y/σ} ( 1-μ)^{(n-y)/σ}/ y^{y/σ} ( n-y)^{(n-y)/σ}]

for y=0,1,2, ...,Inf, μ>0 and σ>0 where C is the constant of proportinality which is calculated numerically using the function `GetBI_C()`.

## Value

The function `DBI` returns a `gamlss.family` object which can be used to fit a double binomial distribution in the `gamlss()` function.

## Author(s)

Mikis Stasinopoulos, Bob Rigby, Marco Enea and Fernanda de Bastiani

## References

Efron, B., 1986. Double exponential families and their use in generalized linear Regression. Journal of the American Statistical Association 81 (395), 709-721.

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. (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.

`BI`,`BB`
 ```1 2 3 4 5 6 7 8``` ```DBI() x <- 0:20 # underdispersed DBI plot(x, dDBI(x, mu=.5, sigma=.2, bd=20), type="h", col="green", lwd=2) # binomial lines(x+0.1, dDBI(x, mu=.5, sigma=1, bd=20), type="h", col="black", lwd=2) # overdispersed DBI lines(x+.2, dDBI(x, mu=.5, sigma=2, bd=20), type="h", col="red",lwd=2) ```