NBF: Negative Binomial Family distribution for fitting a GAMLSS

In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

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.

The functions dZINBF, pZINBF, qZINBF and rZINBF define the density, distribution function, quantile function and random generation for the zero inflated negative binomial family, ZINBF(), distribution a four parameter distribution.

Usage

NBF(mu.link = "log", sigma.link = "log", nu.link = "log")

dNBF(x, mu = 1, sigma = 1, nu = 2, log = FALSE)

pNBF(q, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

qNBF(p, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

rNBF(n, mu = 1, sigma = 1, nu = 2)

ZINBF(mu.link = "log", sigma.link = "log", nu.link = "log", 
      tau.link = "logit")
      
dZINBF(x, mu = 1, sigma = 1, nu = 2, tau = 0.1, log = FALSE)

pZINBF(q, mu = 1, sigma = 1, nu = 2, tau = 0.1, lower.tail = TRUE, 
      log.p = FALSE)
      
qZINBF(p, mu = 1, sigma = 1, nu = 2, tau = 0.1, lower.tail = TRUE, 
      log.p = FALSE)      
      
rZINBF(n, mu = 1, sigma = 1, nu = 2, tau = 0.1)      

Arguments

mu.link	The link function for mu
sigma.link	The link function for sigma
nu.link	The link function for nu
tau.link	The link function for tau
x	vector of (non-negative integer)
mu	vector of positive means
sigma	vector of positive dispersion parameter
nu	vector of power parameter
tau	vector of inflation 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

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 distribution has mean \mu and variance \mu+\sigma \mu^{\nu}. For more details see pp 507-508 of Rigby et al. (2019).

The zero inflated negative binomial family ZINBF is defined as an inflated at zero NBF.

Value

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

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Anscombe, F. J. (1950) Sampling theory of the negative binomial and logarithmic distributions, 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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/9780429298547")}. An older version can be found in https://www.gamlss.com/.

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

(see also https://www.gamlss.com/).

See Also

NBI, NBII

Examples

NBF() # default link functions 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')
# zero inflated NBF
ZINBF() # default link functions  for the zero inflated NBF 
# plotting the distribution
plot(function(y) dZINBF(y, mu = 10, sigma = 0.5, nu=2, tau=.1 ), 
     from=0, to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rZINBF(1000, mu=5, sigma=0.5, nu=2, tau=0.1))
r <- barplot(tN, col='lightblue')
## Not run: 
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]

## End(Not run)

mstasinopoulos/GAMLSS-Distibutions documentation built on June 11, 2025, 7:17 p.m.