Estimates the two parameters of an inverse binomial distribution by maximum likelihood estimation.
Link function for the rho and lambda parameters.
Numeric. Optional initial values for rho and lambda.
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.
An object of class
The object is used by modelling functions such as
This VGAM family function only works reasonably well with
Good initial values are needed; if convergence failure occurs use
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
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
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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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