betageometric: Beta-geometric Distribution Family Function

Description

Maximum likelihood estimation for the beta-geometric distribution.

Usage

betageometric(lprob = "logit", lshape = "loge",
              iprob = NULL,    ishape = 0.1,
              moreSummation=c(2,100), tolerance=1.0e-10, zero=NULL)

Arguments

lprob, lshape

Parameter link functions applied to the parameters $p$ and $\phi$ (called prob and shape below). The former lies in the unit interval and the latter is positive. See Links fo

iprob, ishape

Numeric. Initial values for the two parameters. A NULL means a value is computed internally.

moreSummation

Integer, of length 2. When computing the expected information matrix a series summation from 0 to moreSummation[1]*max(y)+moreSummation[2] is made, in which the upper limit is an approximation to infinity. Here, y is the

tolerance

Positive numeric. When all terms are less than this then the series is deemed to have converged.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. If used, the value must be from the set {1,2}.

Value

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

Details

A random variable $Y$ has a 2-parameter beta-geometric distribution if $P(Y=y) = p (1-p)^y$ for $y=0,1,2,\ldots$ where $p$ are generated from a standard beta distribution with shape parameters shape1 and shape2. The parameterization here is to focus on the parameters $p$ and $\phi = 1/(shape1+shape2)$, where $\phi$ is shape. The default link functions for these ensure that the appropriate range of the parameters is maintained. The mean of $Y$ is $E(Y) = shape2 / (shape1-1) = (1-p) / (p-\phi)$.

The geometric distribution is a special case of the beta-geometric distribution with $\phi=0$ (see geometric). However, fitting data from a geometric distribution may result in numerical problems because the estimate of $\log(\phi)$ will 'converge' to -Inf.

References

Paul, S. R. (2005) Testing goodness of fit of the geometric distribution: an application to human fecundability data. Journal of Modern Applied Statistical Methods, 4, 425--433.

Examples

Run this code

bg.data = data.frame(y = 0:11, wts = c(227,123,72,42,21,31,11,14,6,4,7,28))
fit  = vglm(y ~ 1, betageometric, bg.data, weight = wts, trace = TRUE)
fitg = vglm(y ~ 1,     geometric, bg.data, weight = wts, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
sqrt(diag(vcov(fit, untransform = TRUE)))
fit@misc$shape1
fit@misc$shape2
# Very strong evidence of a beta-geometric:
pchisq(2*(logLik(fit) - logLik(fitg)), df = 1, lower.tail = FALSE)

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