NBII: Negative Binomial type II distribution for fitting a GAMLSS

In Stan125/gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

Description

The NBII() function defines the Negative Binomial type II distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dNBII, pNBII, qNBII and rNBII define the density, distribution function, quantile function and random generation for the Negative Binomial type II, NBII(), distribution.

Usage

NBII(mu.link = "log", sigma.link = "log")
dNBII(x, mu = 1, sigma = 1, log = FALSE)
pNBII(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qNBII(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rNBII(n, mu = 1, sigma = 1)

Arguments

mu.link	Defines the mu.link, with "log" link as the default for the mu parameter
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter
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]

Details

Definition file for Negative Binomial type II distribution.

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

for y=0,1,2, ...,Inf, μ>0 and σ>0. This parameterization was used by Evans (1953) and also by Johnson et al. (1993) p 200.

Value

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

Note

mu is the mean and ((1+sigma)*mu)**0.5 is the standard deviation of the Negative Binomial type II distribution, so sigma is a dispersion parameter

Author(s)

Mikis Stasinopoulos [e-mail], Bob Rigby and Calliope Akantziliotou

References

Evans, D. A. (1953). Experimental evidence concerning contagious distributions in ecology. Biometrika, 40: 186-211.

Johnson, N. L., Kotz, S. and Kemp, A. W. (1993). Univariate Discrete Distributions, 2nd edn. Wiley, New York.

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.

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, NBI, PIG, SI

Examples

NBII()  # gives information about the default links for the Negative Binomial type II distribution  
# plotting the distribution
plot(function(y) dNBII(y, mu = 10, sigma = 0.5 ), from=0, to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rNBII(1000, mu=5, sigma=0.5))
r <- barplot(tN, col='lightblue')
# library(gamlss)
# data(aids)
# h<-gamlss(y~cs(x,df=7)+qrt, family=NBII, data=aids) # fits a model 
# plot(h)
# pdf.plot(family=NBII, mu=10, sigma=0.5, min=0, max=40, step=1)

Stan125/gamlss.dist documentation built on May 12, 2019, 7:38 a.m.