NBI: Negative Binomial type I distribution for fitting a GAMLSS

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

Negative Binomial type I distribution for fitting a GAMLSS

Description

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.

Usage

NBI(mu.link = "log", sigma.link = "log")
dNBI(x, mu = 1, sigma = 1, log = FALSE)
pNBI(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qNBI(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rNBI(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 I distribution.

P(Y=y|\mu, \sigma)= \frac{\Gamma(y+\frac{1}{\sigma})}{\Gamma(\frac{1}{\sigma}) \Gamma(y+1)}\hspace{1mm}\left( \frac{\sigma \mu}{1+\sigma \mu}\right)^y \hspace{1mm}\left( \frac{1}{1+\sigma \mu} \right)^{1/\sigma}

for y=0,1,2,\ldots,\infty, \mu>0 and \sigma>0. This parameterization is equivalent to that used by Anscombe (1950) except he used \alpha=1/\sigma instead of \sigma, see also pp. 483-485 of Rigby et al. (2019).

Value

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

Warning

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

Note

\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)

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

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.

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. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

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

See Also

gamlss.family, NBII, PIG, SI

Examples

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)

gamlss.dist documentation built on Aug. 24, 2023, 1:06 a.m.