gamlss.dist (version 5.0-0)

NBF: Negative Binomial Family distribution for fitting a GAMLSS

Description

The NBF() function defines the Negative Binomial family distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dNBF, pNBF, qNBF and rNBF define the density, distribution function, quantile function and random generation for the Negative Binomial family, NBF(), distribution

Usage

NBF(mu.link = "log", sigma.link = "log", nu.link = "identity")
dNBF(x, mu = 0, sigma = 1, nu = 2, log = FALSE)
pNBF(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
qNBF(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
rNBF(n, mu = 0, sigma = 1, nu = 2)

Arguments

mu.link

The link function for mu

sigma.link

The link function for sigma

nu.link

The link function for nu

x

vector of (non-negative integer)

mu

vector of positive means

sigma

vector of positive despersion parameter

nu

vector of power parameter

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]

p

vector of probabilities

q

vector of quantiles

n

number of random values to return

Value

returns a gamlss.family object which can be used to fit a Negative Binomial Family distribution in the gamlss() function.

Details

The definition for Negative Binomial Family distribution , NBF, is similar to the Negative Binomial type I. The probability function of the NBF can be obtained by replacing \(\sigma\) with \(\sigma \mu^{\nu-2}\) where \(\nu\) is a power parameter. The distrobution has mean \(\mu\) and variance \(\mu+\sigma \mu^{vu}.\)

References

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.

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.

See Also

NBI, NBII

Examples

NBF()   # gives information about the default links for the Negative Binomial Family 
# plotting the distribution
plot(function(y) dNBF(y, mu = 10, sigma = 0.5, nu=2 ), from=0, to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rNBF(1000, mu=5, sigma=0.5, nu=2))
r <- barplot(tN, col='lightblue')

library(gamlss)
data(species)
species <- transform(species, x=log(lake))
m6 <- gamlss(fish~poly(x,2), sigma.fo=~1, data=species, family=NBF, n.cyc=200)
fitted(m6, "nu")[1]