Inverse Binomial Distribution Family Function

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.

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

Author

T. W. Yee

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.

Examples

idata <- data.frame(y = rnbinom(n <- 1000, mu = exp(3), size = exp(1)))
fit <- vglm(y ~ 1, inv.binomial, data = idata, trace = TRUE)
#> Iteration 1: loglikelihood = -246318.9
#> Iteration 2: loglikelihood = -245324.08
#> Iteration 3: loglikelihood = -244341.44
#> Iteration 4: loglikelihood = -243391.08
#> Iteration 5: loglikelihood = -242522.41
#> Iteration 6: loglikelihood = -241837.76
#> Iteration 7: loglikelihood = -241464.45
#> Iteration 8: loglikelihood = -241375.16
#> Iteration 9: loglikelihood = -241370.96
#> Iteration 10: loglikelihood = -241370.95
#> Iteration 11: loglikelihood = -241370.95
with(idata, c(mean(y), head(fitted(fit), 1)))
#> [1] 20.291 20.291
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = inv.binomial, data = idata, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1 -5.64687    1.41462  -3.992 6.56e-05 ***
#> (Intercept):2 -1.94354    0.03156 -61.583  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: extlogitlink(rho, min = 0.5, max = 1), 
#> loglink(lambda)
#> 
#> Log-likelihood: -241371 on 1998 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 11 
#> 
coef(fit, matrix = TRUE)
#>             extlogitlink(rho, min = 0.5, max = 1) loglink(lambda)
#> (Intercept)                             -5.646867       -1.943542
Coef(fit)
#>       rho    lambda 
#> 0.5017581 0.1431958 
sum(weights(fit))  # Sum of the prior weights
#> [1] 1000
sum(weights(fit, type = "work"))  # Sum of the working weights
#> [1] 1004.513