gamlss.dist (version 6.1-11)

NBII: Negative Binomial type II distribution for fitting a GAMLSS

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)

Value

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

Details

Definition file for Negative Binomial type II distribution. $$P(Y=y|\mu,\sigma)= \frac{\Gamma(y+\frac{\mu}{\sigma}) \sigma^y }{\Gamma(\frac{\mu}{\sigma})\Gamma(y+1) (1+\sigma)^{y+\mu/\sigma}}$$ for \(y=0,1,2,...,\infty\), \(\mu>0\) and \(\sigma>0\). This parameterization was used by Evans (1953) and also by Johnson et al. (1993) p 200, see also pp. 485-487 of Rigby et al. (2019). Note that in this parameterization \(E(y)=\mu\) and \(Var(y)=\mu(1+\sigma\mu)\).

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.

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, tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://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, tools:::Rd_expr_doi("10.18637/jss.v023.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. tools:::Rd_expr_doi("10.1201/b21973")

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

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)