Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

The `NBI()`

function defines the Negative Binomial type I distribution, a two parameter distribution, for a `gamlss.family`

object to be used
in GAMLSS fitting using the function `gamlss()`

.
The functions `dNBI`

, `pNBI`

, `qNBI`

and `rNBI`

define the density, distribution function, quantile function and random
generation for the Negative Binomial type I, `NBI()`

, distribution.

1 2 3 4 5 |

`mu.link` |
Defines the |

`sigma.link` |
Defines the |

`x` |
vector of (non-negative integer) quantiles |

`mu` |
vector of positive means |

`sigma` |
vector of positive despersion parameter |

`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] |

Definition file for Negative Binomial type I distribution.

*P(Y=y|μ, σ)=
Γ(y+1/σ)/Γ(1/σ) Γ(y+1) ((σ μ)/ (1+σ
μ))^y(1/(1+σ μ))^{1/σ}*

for *y=0,1,2, ...,Inf*, *μ>0* and *σ>0*. This
parameterization is equivalent to that used by Anscombe (1950) except he used *alpha=1/sigma* instead of *sigma*.

returns a `gamlss.family`

object which can be used to fit a Negative Binomial type I distribution in the `gamlss()`

function.

For values of *sigma<0.0001* the d,p,q,r functions switch to the Poisson distribution

*mu* is the mean and *(mu+sigma*mu^2)^0.5* is the standard deviation of the Negative Binomial
type I distribution (so *sigma* is the dispersion parameter in the usual GLM for the negative binomial type I distribution)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

Anscombe, F. J. (1950) Sampling theory of the negative bimomial and logarithmic distributiona, *Biometrika*, **37**, 358-382.

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 http://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, 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`

, `NBII`

, `PIG`

, `SI`

1 2 3 4 5 6 7 8 9 10 11 | ```
NBI() # gives information about the default links for the Negative Binomial type I distribution
# plotting the distribution
plot(function(y) dNBI(y, mu = 10, sigma = 0.5 ), from=0, to=40, n=40+1, type="h")
# creating random variables and plot them
tN <- table(Ni <- rNBI(1000, mu=5, sigma=0.5))
r <- barplot(tN, col='lightblue')
# library(gamlss)
# data(aids)
# h<-gamlss(y~cs(x,df=7)+qrt, family=NBI, data=aids) # fits the model
# plot(h)
# pdf.plot(family=NBI, mu=10, sigma=0.5, min=0, max=40, step=1)
``` |

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