inv.binomial {VGAM}

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.

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.

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.

Author(s)

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

See Also

negbinomial, poissonff.

Examples

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