VGAM (version 1.0-1)

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 $E(Y) = shape2 / (shape1-1) = (1-p) / (p-\phi)$ if shape1 > 1, and if so, then this is returned as the fitted values.

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.

See Also

geometric, betaff, rbetageom.

Examples

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