Negative Binomial type I distribution for fitting a GAMLSS
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.
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)
mu.link |
Defines the |
sigma.link |
Defines the |
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] |
Definition file for Negative Binomial type I distribution.
P(Y=y|μ, σ)= Γ(y+1/σ)/Γ(1/σ) Γ(y+1) ((σ μ)/ (1+σ μ))^y(1/(1+σ μ))^{1/σ}
for y=0,1,2, ...,Inf, μ>0 and σ>0. This parameterization is equivalent to that used by Anscombe (1950) except he used alpha=1/sigma instead of sigma.
returns a gamlss.family object which can be used to fit a Negative Binomial type I distribution in the gamlss() function.
For values of sigma<0.0001 the d,p,q,r functions switch to the Poisson distribution
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)
Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou
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. 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, https://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.
gamlss.family, NBII, PIG, SI
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)
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