# inv.binomial: Inverse Binomial Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

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

## Usage

 ```1 2``` ```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`.

## Details

The inverse binomial distribution of Yanagimoto (1989) has density function

f(y;rho,lambda) = (lambda * Gamma(2y+lambda)) * [rho*(1-rho)]^y * rho^lambda / (Gamma(y+1) * Gamma(y+lambda+1))

where y=0,1,2,... and 0.5 < rho < 1, and lambda > 0. The first two moments exist for rho>0.5; 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.

## Value

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

## Note

This VGAM family function only works reasonably well with intercept-only models. Good initial values are needed; if convergence failure occurs use `irho` and/or `ilambda`.

Some elements of the working weight matrices use the expected information matrix while other elements use the observed information matrix. Yet to do: using the mean and the reciprocal of lambda results in an EIM that is diagonal.

T. W. Yee

## 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`.
 ```1 2 3 4 5 6 7 8``` ```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 ```