VGAM (version 1.1-6)

# inv.binomial: Inverse Binomial Distribution Family Function

## Description

Estimates the two parameters of an inverse binomial distribution by maximum likelihood estimation.

## Usage

inv.binomial(lrho = extlogitlink(min = 0.5, max = 1),
llambda = "loglink", irho = NULL, ilambda = NULL, zero = NULL)

## Arguments

lrho, llambda

Link function for the $$\rho$$ and $$\lambda$$ parameters. See Links for more choices.

irho, ilambda

Numeric. Optional initial values for $$\rho$$ and $$\lambda$$.

zero

See CommonVGAMffArguments.

## Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

## Details

The inverse binomial distribution of Yanagimoto (1989) has density function $$f(y;\rho,\lambda) = \frac{ \lambda \,\Gamma(2y+\lambda) }{\Gamma(y+1) \, \Gamma(y+\lambda+1) } \{ \rho(1-\rho) \}^y \rho^{\lambda}$$ where $$y=0,1,2,\ldots$$ and $$\frac12 < \rho < 1$$, and $$\lambda > 0$$. The first two moments exist for $$\rho>\frac12$$; then the mean is $$\lambda (1-\rho) /(2 \rho-1)$$ (returned as the fitted values) and the variance is $$\lambda \rho (1-\rho) /(2 \rho-1)^3$$. The inverse binomial distribution is a special case of the generalized negative binomial distribution of Jain and Consul (1971). It holds that $$Var(Y) > E(Y)$$ so that the inverse binomial distribution is overdispersed compared with the Poisson distribution.

## References

Yanagimoto, T. (1989). The inverse binomial distribution as a statistical model. Communications in Statistics: Theory and Methods, 18, 3625--3633.

Jain, G. C. and Consul, P. C. (1971). A generalized negative binomial distribution. SIAM Journal on Applied Mathematics, 21, 501--513.

Jorgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall

negbinomial, poissonff.

## Examples

Run this code
# NOT RUN {
idata <- data.frame(y = rnbinom(n <- 1000, mu = exp(3), size = exp(1)))
fit <- vglm(y ~ 1, inv.binomial, data = idata, trace = TRUE)
with(idata, c(mean(y), head(fitted(fit), 1)))
summary(fit)
coef(fit, matrix = TRUE)
Coef(fit)
sum(weights(fit))  # Sum of the prior weights
sum(weights(fit, type = "work"))  # Sum of the working weights
# }


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